£. b.C. 1*2
TRANSACTIONS
ROYAL SOCIETY OF EDINBURGH
TRANSACTIONS
OF THE
ROYAL SOCIETY
OF
EDINBURGH.
VOL. XLI.
EDINBURGH:
PUBLISHED BY BOBEET GEANT & SON, 107 PRINCES STREET.
AND WILLIAMS & NOBGATE, 14 HENEIETTA STEEET, COVENT GAEDEN, LONDON.
MDCCCCVI.
No.
I.
Published
February 5, 1904.
No.
XVIII.
II.
)>
February 16, 1904.
>>
XIX.
III.
>»
March 3, 1904.
>i
XX.
IV.
!)
April 9, 1904.
)>
XXI.
V.
))
May 4, 1904.
>>
XXII.
VI.
)>
May 27, 1904.
>)
XXIII.
VII.
n
July 1, 1904.
j)
XXIA r .
VIII.
>j
August 5, 1904.
»
XXV.
IX.
>)
September 2, 1904.
>!
XXVI.
X.
>)
September 9, 1904.
>>
XXVII.
XI.
jj
November 19, 1904.
)!
XXVIII.
XII.
:>
December 14, 1904.
,,
XXIX.
XIII.
>)
January 13, 1905.
) J
XXX.
XIV.
))
June 9, 1905.
))
XXXI.
XV.
it
March 3, 1905.
n
XXXII
XVI.
jl
April 15, 1905.
^»
XXXIII
XVII.
,.
April 18 ; 1905.
Published
May 26, 1905.
May 6, 1905.
May 13, 1905.
June 15, 1905.
June 9, 1905.
June 30, 1905.
July 3, 1905.
July 3, 1905.
July 31, 1905.
July 20, 1905.
August 7, 1905.
August 30, 1905.
September 25, 1905.
November S, 1905.
November 7, 1905.
January 18, 1906.
CONTENTS.
PART I. (190304.)
NUMBER
I. On Generalised Functions of Legendre and Bessel. By the Rev. F. H
Jackson, H. M.S. " Irresistible," ....
II. Certain Fundamental Power Series and their Differential Equations,
By the Rev. F. H. Jackson, H.M.S. " Irresistible,"
III. Magnetization and Resistance of Nickel Wire at High Temperatures,
By Professor C. G. Knott, D.Sc,
IV. The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D
(Edin.), M.A. (Camb.), F.G.S., Lecturer in Geology at the University of
St Andrews. (With a Plate), .....
V. Spectroscopic Observations of the Potation of the Sun. By Dr J. Halm,
Assistant Astronomer at the Royal Observatory, and Lecturer in
Astronomy at the University, Edinburgh, . . . .89
VI. Theorems relating to a Generalisation of the Bessel Function. By the
Rev. F. H. Jackson, H.M.S. " Irresistible," . . . .105
VII. On Some Points in the Early Development of Motor Nerve Trunks and
Myotomes in Lepidosiren paradoxa (Fitz.). By J. Graham Kerr,
Regius Professor of Zoology in the University of Glasgow, . . 119
VIII. An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate.
By John Dougall, M.A., . . . . . .129
29
39
53
PART II. (190405.)
IX. On the Measurement of Stress by Thermal Methods, ivith an Account of
some Experiments on the Influence of Stress on the Thermal Expan
sion of Metals. By E. G. Coker, M.A. (Cantab.), D.Sc. (Edin.), F.R.S.E.,
Assistant Professor of Civil Engineering, M'Gill University, Montreal.
(With Two Plates), ....... 229
vi CONTENTS.
PAGE
X. On the Spectrum of Nova Persei and the Structure of its Bands, as
photographed at Glasgow. By L. Becker, Ph.D., Professor of
Astronomy in the University of Glasgow. (With Three Plates), . 251
XI. The Histology of the Blood of the Larva of Lepidosiren paradoxa.
Part I. Structure of the Besting and Dividing Corpuscles. By
Thomas H. Bryce, M.A., M.D. (With Five Plates), . . .291
XII. The Action of Chloroform upon the Heart and Arteries. By E. A.
Schafkr, F.R.S., andH. J. Scharlieb, M.D., C.M.G., . . 311
XIII. Continuants resolvable into Linear Factors. By Thomas Muir, LL.D., 343
XIV. The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D.
Falconer, M.A., B.Sc. (With a Map), .... 359
XV. On a New Family and Twelve New Species of Botifera of the Order
Bdelloida, collected by the Lake Survey. By James Murray. (With
Seven Plates), ........ 367
XVI. The Eliminant of a Set of General Ternary Quadrics. — (Part III.)
By Thomas Muir, LL.D., . . . . . .387
XVII. Theorems relating to a Generalization of BesseVs Function. By the
Rev. F. H. Jackson, R.N., . . . . . .399
XVIII. On Pennella balcenopterce: a Crustacean, parasitic on a Finner Whale,
Balsenoptera musculus. By Sir William Turner, K.C.B., D.C.L.,
F.R.S. (With Four Plates), ...... 409
XIX. The Histology of the Blood of the Larva of Lepidosiren paradoxa.
Part II. — Hcematogenesis. By Thomas H. Bryce, M.A., M.D. (With
Four Plates), ........ 435
XX. Supplement to the Lower Devonian Fishes of Gemiinden. By R. H.
Traquair, M.D., LL.D., F.R.S. (With Three Plates), . . 469
PART III. (190405.)
XXI. A further Contribution to the Freshwater Plankton of the Scottish
Lochs. By W. West, F.L.S., and G. S. West, M.A., F.L.S. (With
Seven Plates), . . . . . . . .477
XXII. The Nudibranchiata of the Scottish National Antarctic Expedition.
I'.v Sir Charles Eliot, K.C.M.G., . . . . .519
CONTENTS. Vll
NUMBER PAGE
XXIII. On the Internal Structure of Sigillaria elegans of Brongniart's
" Histoire des vegetaux fossiles." By Robert Kidston, F.R.S.L. &
E., F.G.S. (With Three Plates), . . . .' .533
XXIV. On the Structure of the Series of Line and BandSpectra. By
J. Halm, Ph.D., ....... 551
XXV. On the Hydrodynamical Theory of Seiches. By Professor Chrystal.
With a Bibliographical Sketch, . . . . .599
XXVI. On a Group of Linear Differential Equations of the 2nd Order,
including Professor ChrystaVs Seiche equations. By J. Halm,
Ph.D., ........ 651
XXVII. The Tardigrada of the Scottish Lochs. By James Murray. (With
Four Plates), ....... 677
XXVIII. The Plant Remains in the Scottish Peat Mosses. Part I. — The
Scottish Southern Uplands. By Francis J. Lewis, F.L.S. (With
Six Plates), ....... 699
XXIX. Semiregular Networks of the Plane in Absolute Geometry. By
Duncan M. Y. Sommerville, M.A., B.Sc. (With Twelve Plates),. 725
XXX. A Monograph on the general Morphology of the Myxinoid Fishes,
based on a study of Myxine. Part I. — The Anatomy of the
Skeleton. By Frank J. Cole, B.Sc. Oxon. (With Three Plates), . 749
XXXI. The LifeHistory of Xenopus lsevis, Daud. By Edward J. Bles,
B.A., B.Sc, Assistant in Zoology at the University of Glasgow.
(With Four Plates), . . . . . .789
XXXII. Calculation of the Periods and Nodes of Lochs Earn and Treig,
from the Bathymetric Data of the Scottish Lake Survey. By
Professor Chrystal and Ernest MaclaganWedderburn, M.A.
(With Two Maps), ....... 823
XXXIII. The Alcyonarians of the Scottish National Antarctic Expedition.
By Professor J. Arthur Thomson, M.A., and Mr James Ritchie,
M.A. (With Two Plates), . . . . . .851
Appendix —
The Council of the Society, . . . . . .865
Alphabetical List of the Ordinary Fellows, . . . .867
List of Honorary Fellows, . . . . . .886
List of Ordinary and Honorary Felloivs Elected during Session 1904
1905, ........ 888
viii CONTENTS.
41 f£B.19D7
PAGK
Fellows Deceased, 190405, . . . . . .889
Laws of the Society, ....... 893
The Keith, Mahdougall Brisbane, Neill, and Gunning Victoria Jubilee
Prizes, 899
Awards of the Keith, MakdougallBrisbane, and Neill Prizes from 1827
to 1904, and of the Gunning Victoria Jubilee Prize from 1884 to
1904, 002
Proceedings of the Statutory General Meeting, 1904, . . .911
Index, .......... 913
". tf?
TRANSACTIONS
OF THE
ROYAL SOCIETY OF EDINBURGH.
VOLUME XLI. PART I.— FOR THE SESSION 19034.
CONTENTS.
I. On Generalised Functions of Legendre and Bessel. By the Rev. F. H. Jackson, H.M.S.
"Irresistible," ..........
(Issued separately 5th February 1904)
II. Certain Fundamental Power Series and their Differential Equation*. By the Rev. F. H.
Jackson, H.M.S. " Irresistible," .......
(Issued separately 1 6th February 190 %.)
III. Magnetization and Resistance of Nickel Wire at High Temperatures. By Prof. C. G.
Knott, D.Sc, ..........
(Issued separately 3rd March 1904.)
IV. The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D. (Edin.), M.A. (Camb.),
F.G.S., Lecturer in Geology at the University of St Andrews. (With a Plate),
(Issued separately 9th April 1901/..)
V. Spectroscopic Observations of the Notation of tlie Sun. By Dr J. Halm, Assistant Astronomer
at the Royal Observatory, and Lecturer in Astronomy at the University, Edinburgh,
(Issued separately Jftli May 190 If.)
VI. Theorems relating to a Generalisation of the BesselFunction By the Rev. F. H. Jackson,
H.M.S. "Irresistible," .........
(Issued separately 27th May 1904.)
VII. On Some Points in the Early Development of Motor Nerve Trunks and, Myotomes in
Lepidosiren paradoxa (Fitz.). By J. Graham Kerr, Regius Professor of Zoology in
the University of Glasgow, ........
{Issued separately 1st July 190 %.)
VIII An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate. By John
Douoall, M.A., .........
{Issued separately 5th August 1904.)
Pace
29
39
53
S9
105
119
129
ERRATUM.
Vol. XLI., Part I., No. VI., p. 105, line 9 from the bottom of page, "multiply  by p***" 1 '
*■ (M)
EDINBURGH:
PUBLLSHED BY ROBERT GRANT & SON. 107 PRINCES STREET
AND WILLIAMS & NORGATE, 14 HENRIETTA STREET, CO VENT GARDEN, LONDON.
MDCCCCIV.
Price Twenty one Shillings.
TEANSACTIONS.
I.— On Generalised Functions of Legendre and Bessel. By the Rev. P. H.
Jackson, H.M.8. " Irresistible." Communicated by Dr W. Peddie.
(Part I. received July 27 ; Part II. August 6, 1903. Read November 16, 1903. Issued separately February 5, 1904.)
Part I.
1.
In this paper some properties of the functions J [n] (x , X) , P [n] (a; , X) , Q in] (x , X) will be
investigated. These functions are generalised forms of Bessel's and Legendre's
Functions. Two interesting expressions are obtained for the sum of the coefficients
of x in the series P M (^) and ( ^ ] . Throughout the paper [n] denotes ± .
J [n] (x , X) denotes the convergent series
X«+2r~[n+2rJ
in which
Up 1 we see that
P [n] (xX) denotes
Z [,■]![« + r]!(2),(2)„ + ,
[n] ! represents [1] [2] [3] . . . . [?i]
(2)„ „ p+lp 2 +l p 3 + l .
[n] ! reduces to n !
(2)* „ 2"
(1)
p n + 1
[2«  2r]\
Zu K , [r\\[nr}\[n2r]\{2) l .{2) n J
;) r.r+2^n2c a .[«2cj
(2)
which when p = 1 reduces to Legendre's function P,
Qm](x , X) denotes
^[r]!> + 2r+l]!(2) r
(3)
If w be not integral then [n]! must be replaced by the function H p ([n]) which is
defined as
■wo  ^Bf.;ii::::::a,/ ^
_j»[»i + 1] [n + 2] [» + 3]
TRANS. ROY. SOG. EDIN., VOL. XLI. PART I. (NO. 1).
(*)
2 THE REV. F. H. JACKSON ON
reducing when y>= 1 to Gauss's expression for n(n) or T(n + l)
I Li? ' 34 * „
r(«+l)  JUn +1 . n + 3 . n + s n + K K
The infinite product (4) is convergent for all values of n except negative integers :
(2)„ will be in general
TT /l\,1\
x( P +\y . . . (5)
n,,(M)
n p (M)
for the infinite product H^([»]) is
j) 2 l p*_~l p 6  1 y' 2K  1
p*  1 i? 2  1 ^ 2 l
which is
(2) n denoting a convergent infinite product reducing for integral values of n to
(/y + l)(p 2 +l) . . . (p n +l).
The difference theorem for Il p ([x]) is n p ([#]) = [afjll^fa;— l]) . The multiplication
theorem which is the generalisation of
T(^T(x + \ ■ ■ ■ T(.<: + n ~\ = T(nx) . (27r)'^n^ nx
is investigated in another paper.
As obtained from the differential equations the series are perfectly general with
regard to n, and the question of [n]! or II([n]) only arises in connection with the
arbitrary constant multiplier of each series.
2.
In a paper on generalised forms of the series of Bessel and Legendre (Proc.
Edin. Math. Soc, vol. xxi.), I have shown that if J [n] (x) denote the series
I x l " +2] a/" +4] )
I xi ' li+ [2][2n + 2] + [2][4][2n + 2][2n+4] + ' ' " " J ' " (6)
then y = J (n ](^) satisfies a differential equation
ptfQj + ! 1  [n]  [  n]}x d £ + [»] [  n]y = iW^ . . (7)
(/"' , d d
— — ■ denoting — —  . —
dz m ° d(x p ) dx
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 3
Introducing a parameter X we see that (6) and (7) may be stated more generally as
_ , \ w X 2 z [ " +2] XV+* 1 I ,
V ~ \ X + [2][2» + 2] + [2][4][2« + 2][>T+] ' ' ' ' f ' W
and
^S + { 1 ~ W [»]} ^ + M [  »]» = *W M (^) ■ 0)
A"
In the series (8) give to the arbitrary constant A the value or generally
X n
(2) II(Tr?1V ^ en throughout the paper J [n] (cc,X) will denote the series
A." r=co A 2 'a; [ " + "' ]
2,r2U4l . . . l2rl.2M + 2l r2ra + 2rl ' ' ^ 0)
(2)»W! J6?[2][4] • • • [2r],[2n + 2] [2 ? z + 2r]
which can be written in the form
2(2) r (2) B+r [7]![» + r]! ' (11)
Since
[21 [4] . . . [2r] = t 1 . ti i^ 1
L JL J L J p _ 1 p _j p _ 1
r r £ p1 p  I p  1
= (2)rW!
and in general (r not integral)
(^ + i)'n, 2 ([7]) = (2),.n,([r])
3.
It may be verified at once that
jj [0] (*A)j = AJ ra (afA) . . . (12)
d_
dx
and in general that
for
^s } J ajwj^X) } = AJ r „ +1] (a^ • X) . . . (13)
ar w J w (sX)
(2) r (2)„ +r [r]! [» + »■]!
^fHJr a) p™[2r]
<(2) r (2)„ +r [r]! [» + »•]!
A. B + 2r ajp"[ 2 '"J
= ^(2).C2) Ml [n + r\[ ' ' ^
since
[n + 2r][n] = L 1 _^_i = p»^_i = v "[2r]
p — 1 p1 p1
4 THE REV. F. H. JACKSON ON
Differentiating the series (14) with regard to x p " we see that the differential
coefficient of the first term is zero and we obtain the series
V A""[2r>  " , (15)
^(2),.(2) B+r [r]![« + r]!
ra ^n+'Jr/gDtnJjjrtn+Srl]
= Z(2) r (2) B+r [rlj!L« + rJ!
= Xa:_Mnl 2(2W2),, +1+r [r]![n + l+r]!
= Aa*»)J [B+1] (ajPX)
which establishes
^•")^ H " J " l(a ' A) ^ = Xx ~ vl " ]J ^^"^ • • ' (16)
By a change of the independent variable this may be written in the form
from which we have by repeating the operations
A „ +1) ,_ / ,« + . t „ +11J)+ii( ^. + . A) =
1 _A J_L '' J . 1 rf ( 1 rfJ„(a:A)  ) »i»
.w ;J " +1 ,/(,/") I .^"^' ( /(, J "') 1 f a 3 d(x"') \ *" dx U" ' i ] ] [ )
a theorem analogous to
4.
Similarly we may easily verify that
\x"J U)] (x"'\) = J I xJ (1] (^A) } . . . (19)
and in general that
X,^J ln . Yi (x^ + \) =   ^J M (o* n A) j . . (20)
from which by repeating the operation
lii the case p = 1, this becomes
,V « ',U„ J x k) = (2 +1  .r«+'J, 1+ .,(*,A) }
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL.
5.
A very symmetrical differential equation is
To prove this we must first show that
J t ««(^)J[» + i](^) = ^?U^P) • • ■ (23)
The coefficient of x pln+2r ' 1] in J^. v (af\)  J^aA) is
\n+2rl \n+2rl
[ r ]! [ n + r 1]! (2),(2)„ + ,_ 1 [r  1]! [n + r]\ (2) r _ 1 (2)„ +
which is
k n+2r . p'{2n]
~ [r]![« + r]!(2) r (2), !+ ,A
The r + 1 th term of the resulting series is therefore
[2W] \n+2r p n+ 2r xV[ n+2rll
which since
l
(24)
a; p[H+2rll _ _* _ jgCn+Zr]
a;
shows us that the series is
[ ij M (.>,w  ■ • W
and we have established the relation (23) between three successive functions, reducing
when p = 1 to
J„ 1 J„ + i = J, .... (26)
X
Now it has been shown that
Ax''J [ , 1 _ 11 (^ +, A) = ^.^J [ „ ] (^''A) . . . (27)
Changing the independent variable x to x p " we obtain
Xx^'^J^^X) = ^ { ^"i<'U [H] (*A) } . . . (28)
which is
W^) =^""^7^) { ^ B[ %^) } • • ( 29 >
as can easily be verified independently.
We have already shown that
w**) = x*' 4 'V(x>) \ xA ~ n ' J ^ x ^ \ ■ •  (30 >
6 THE REV. F. H. JACKSON ON
Taking (30) from (29) we obtain by (23) that
The expression on the right is equal to
4 C T*C*> X!^d(^ {'•**)} ■ ' ■ <3l>
Now
is equal to
\p n x im '
and therefore we have finally
[ ^ { *«("*) " Jm(* ' V) } = <^«W**)  ^' n ^fm(^) ■ ( 32 >
a very symmetrical equation.
From (23) it follows that
j ln _^x) = _L  [a»jr.(«xp) +i>«[> + 4] j, 1+ ,(.^) + } . (33>
6.
The Function Y m (x\) .
If P ( „](#) denote the series
A J ^M  MD'" 1 ] „3,.[»2] . [nJ[nl]>2]>3] „ [a _ 4] _ .... I (U \
Then * P[»](#) satisfies the differential equation
**&  3 + { x  w  1  »  1] } * d £ + w [  »  1]// = pU)  p Uo ( 35 >
Introducing a constant parameter X we have more generally the series
I [2][2»l] i [2][4][2nl][2n2]r i K '
Satisfying
■Of  p S + j i  m [■  1] } 4E +'w[— >i"  p u*a>  n,,,c*) <37>
Give to the arbitrary constant A the value
[2n]i A"
(2)»[»]l [«]!
* Proc. /Jdtn. Ma(/i. Soc, vol. xxi.
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL.
Then throughout subsequent work the function P w (a;,A) will denote
<  ')' [»  rJI \JXm (2)„_,(2)/ ^'^" ■ m
J* 1 (x\) denotes the series
r=co T9i» — Strll
ZC  IV" LIZ ILL \ «2»'2„[«2r2] / OQ\
J ; [«r]![r,2r2]![r]!(2) n _,.(2) r A X " ' < 39 >
We proceed to show that
d ■ P [w+1] (a\) _ dP c n x] (g\) = ^r 2a + 1 ]p (^,Ap) . . (40)
(/a: aa; js"
In P 1)l+ i](xX) +fP [n _i](a;X) the terms involving aj [n_8r+1] give us
OT r.r+2/ _ 1 \t [2n 2r + 2]! yn+lir^n+lirl
* ^ '[nr + l]l[»ar+l]![r]I(2) r <2Wir
i ,.„rl.r+l/ _ 1 \rl [2w  li'J. \«+l2r,[m2r+l]
^ l ^ [»r]![«2r+l]![/l]!(2) r _ 1 (2) 7l _ 2 ,
which gives us an expression for the coefficient of x [n ~ 2r+1] , viz.,
(1
]V ^ [2n2r]l I [2,;2r + l] ,_,, [2r] ) ^
J [rer]![«2r]![r]!(2),(2)„_, 1 [n2r+l] ' [ra2r4l]J
2r+l
The part within the large bracket is
p la2r+l_ I  K j,l*r(j]pr  I)
(2) l)[n2r+l]
Putting k= —p this reduces to
^ 2,, [2« + 1]
[»2r+ 1]
Therefore
[2n2r]\p ir [27i+l]
_ X[2w+1] ^, , _ , y  r+2 [2w2?']! xnSr+lUn^r+U
" 2/ }I [nr]\[n2r+l]l[r)\(2) r (2) n _ r
p"
from which we have at once
dP ln+1 jxk)_ dP { „(«*) = W + l]P w (^>p) . . (42)
rte etas j?"
which leads to
P n dVl f* X) = V { O  l]P n i(a: p ,Xp) +p\2n  5]P„_ :< (^Xp) + p°[2n  9]P n _ 5 (a»Wp) + .... J (43)
8.
The relation between three consecutive functions is
[n]P w (*A) = X[2 M l>P [ , 1 _ 1] (^A) i y'+ 1 [nl]P [ , i _. 2] (xA) . . (44)
which corresponds to the ordinary relation
raP„ = (2»l)a;P B _ 1 («l)P B _ 2
^Consider
A[2n  l>P [tl _ 1 ,(ir*A)  K P [n  1]P [;1 _ 2] (^A)
8 THE REV. F. H. JACKSON ON
The terms involving x l "~ 2r] give us
" *< " >r[»  '>' W \»,.. 1J , l ,, [ !2^:lji(2),_,, 1 (2),., X ^^' tf ^'''
These may be written
_ . s [2»  V ]! ___ I ^"^  j>»  p" + 1  Kjr^p'* 1  2 r> j)' 1 + 1 ) ^
I " [» r r]l [»  2r]l [r]!(2)^ r (2)r « ~ (p l) 2 [2n2r 1] ]
Putting *=//' the large bracket reduces to
(p"i)(y"''i)
(pl) a [2ra2rl]
which is
w
so that the coefficient of x {n ' 2r] is
(  1)V*"M [2»2r]l 2r
and the series is
[nJPwCaA)
establishing
[»]P,„ l (asA> = A[2wl>P l „_ 1) (^'A)p' ! + 1 [wl]P [ „_, ] (crA) . . (45)
9.
Another property of the function is
By means of this we establish that
Pm(1P 2 ~) = ^P t »i](lJ?b .... (47)
and if n be integral
Pm(1P 2 1 = /'" • • . . (48)
which corresponds to the theorem that the sum of the coefficients in Legendre's series
is equal to unity. The proof of (46) is as follows :
p,„(xx) . p. l)y^> H , [<t . fJ fc _*$ , tojrf ™ • <«>
therefore
JLtevfr*) \ = Z(l)"/>'" +2 A"'" r[tT J"^* l .. [n2r]^^
tf(z p ) < J o [■'•]! [» r]I[n 2 r]!(2), ( _,(2), L J
Similarly,
(f(af)l I o Lr _lj ![w  r+ l]. Ln _2r+2j!(2)„_ r+1 (2) r _ 1 a!
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 9'
The coefficient of x [n_2r+1] in the expression
XW ^{ P,M(afX) } ~ P Jj\ FU ** X)
d{xP)
is
K }P [r]![»r]![«2r]!(2) r (2)„_ r l L J ' [n  r+ 1] [n  2r + 1] [n  27+ 2](p n ~ r+1 + 1) J
Now since
[nr+l](p n  r+1 + l) = [2ra2r + 2]
the expression within the large bracket reduces to
r , _J"2n2r+l][2r]
[? ' 2r]+i3 [n2r+l]
which is
p~ 2r [w] [n+1]
[n2r+l]
and we may write
«jrf?y**) _ ~^W^ _ r Bl r. + 11 y(_i yyv** O ~ 2r J ! r [»2, +1] / 50y
XX <*(*) * ^ L M J^+1JZ^ l )P k [r]l[nr]l[n2r + l]l(2) r (2) n _,: 1 ^
Now consider
cx? m {x v \)  c 1 P [ „_ 1] (a;A.)
The coefficient of a: 1 " 2 ''"*" 11 in this expression is
C[ l)l> A [r]\[nr]\[n2r]l(2) r (2)„_,. ll ' J [r  1]! [n  r]\ [»  2r + 1]! (2) r _ 1 (2)„_ r
which may be written
+ [r]![«r]![»2r]!(2) r (2) n _ r 1 ' r X? T V' [»2r+l]
ri. r ..], [„ _ 2r ], (2) r (2)„_„ 1 y X> + V [»  2»" + K
Tf now c be chosen as
and c x
the large bracket reduces to
X«[n]
\p[n]
[»][» + !]
[n2r+l]
and we have
^'W*) ^»]P M W  §M [+ 'K*""» [r] , [. _ r] , gViff'l]! (2),(2)„_f "" < M>
and this series has been shown in (50) to be
i
X_2 T m d Vw(x p X) dF [n] (xl>\)
= X>[n] { asP^X) ZFtnipty} .... (52)
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 1). 2
10 THE REV. F. II. JACKSON ON
We see that if X = p h
and X = 1
the left side of the above expression becomes identically zero. Therefore when
A =j> and x= 1
x[»] j,iy ,,(,A) ^Ptn^X) 
must also be zero, which is
P M (l#)=i*P [n _ y (l2>i) . . . (53)
from which we have for integral values of n
P w (li>i)=jP*P ra (li*)
; ; ; ; ; • • • • < 54 )
P w (li>i)=i>*P [w _ 1] (li'4)
and so taking the product of the two sides of the equations
P M (l^)=^Po(l^)=^ • • • (55)
which is
[»]! [«]! (2)„ 1 P [2] [m  1 ] + 1 [2] [4] [2n  1] [2n  3] J * < 00 '
that is 1 ■« + = W'W',^ . . . (57)
[2JL2?ilJ [2ra]! v 7
the general term of the series being
p
.r+i W [ra  1] [n  2] . . . . [n2r + l]
[2] [4] . . [2?] . [2nl] . . . [2«2r+l]
If we put p = 1 this reduces to a wellknown series
, n • n  \ n ■ n  1 • n  2 n 3 w ! n ! 2"
~ 2 • 2m  1 + 2 • 4 • 2n  1 • 2n  3 ~ ' ' " = ~~2toT
10.
xhe series
;/ . A ^_ / ,,[,H[ :: , i i]^,.. ! , + .... . . m
is a solution of a differential equation of the form
/'■''3'A + {l[ re _ v ]_[_ ra _ v _l] }«g + [ii V ] [»» l>/(«)/(«0 (59)
for, assuming that y can be expressed as a convergent or finite series of the form
y = £A^
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 11
performing the operations indicated on the left side of the differential equation we
obtain
{p[m^\ [wij  1] + {1  [re  v]  [  re  v  1]}K] + [re  v] [  n  v  l]}A 1 a*" a  A. 2 [m 1 ][m 1  lJA^™'" 2 !
+ similar expressions in m 2
™ 3 (60)
Since p[ m i] [ n h  ! ] = ["h?  ['"J
the expression which is the coefficient of A^ 1 ™ 11 in (60) reduces to
{H[»"]}{W[»".i]}A 1
so that we have altogether the series
{[toj]  [»v]}{[i»J  [ re v  1]} A^"" 1  X^toJ [«ij  ljA^" 1 ' 23
+ {[m 2 ]>v]}{[> 2 ][>revl]}A^ . (61>
+
Choose m 2 = m 1  2
m 3 = rei 2  2
w r+1 = m,.  2
Also choose
{[m r+1 ]  [re  v]}{[«t,. +1 ]  [  n v  1]}A,. +1 = X. 2 [m r ] [m r .  l]A r
Let w x also be so chosen that the coefficient of x [ntil may vanish, then
m 1 =n — v or re — v — 1
For the value m x =nv
m r+1 = nv2r
we have
a _ a / 4 * 41 [«v2r+2][»v2r+l] ^
A r+1 A r ^ r [2r][2»2r + l] " " ( '
and for the value m x =  re  v  1
we have
m r+1 = rev1 — 2r
a _ a t [re + v + 2rl][re + v + 2r]
A + iA r ^ 2 [ 2r ][2re + 2r+l] ' ' ' ^ 6)
From relation (62) we have the series
» A I ^* 4 =Sftfe^ 3 ® , <" + i • (64)
a solution of the differential equation
^g_X^+{l[rev][ravl]^g + [rev][revl>=/( a; )/K) (65)
f(x) denoting the function
[»,][„„ 1] a { ^^  A n  ^^".7] " 3] £*« + } ( 66 >
12 THE REV. F. H. JACKSON ON
We also have from relation (63)
, A {,>— " + y >+]^"_±J± 2 W^ + } . (67)
a solution of the differential equation in which f(x) denotes
[« + y+l][n + v + 2] A j , „, 3 [» + v + 3] [n + v + 4 ] x . a , tJ _„.,. s] I (6g)
7^^^ A V + ' [2][2n + 3] i
Since \ is quite arbitrary, replace it by \p v and we then have the series
y = const.  A"^"J  [ " ~3& ~J~ l \ *\»*W** + ... } . (69)
a solution of
^iS "W> D " + { * [»"][«" 1]  * + [«  v] [  n  v  1> =/(*) /(^) (70)
The series (69) is when f is integral, and <n
d(*p») w
The series and differential equation are analogous to one given by Heine
(\ x ify2{v+l).r d  y + (n + v+\)(nv)y = () . . (72)
i(.? z ax
of which the primitive is
,A^ + S^ <rc . . (73)
The sum of the series of coefficients in Heine's series
(n v)(nvl ) (n  v)(n  v  1 ){n  v  2)(n  v  3) _
1 " " 22«  1 + ~ 2 • 4 • 2n  1 • 2n  3
is shown in ChrystaVs Algehra, Part II. page 185, or 209 (2nd edition), to be
v\2n\ ' ' • V ;
The analogous theorem in the general series is
, _ ,,+■•[" v][nvl] iv+R [n  v\ [n  v  1] [n  v  2] [n  v  3]
/( [2][2wl] +/ ' [2] [4] O  1] [2n  .8] •••
= (2) n [n]l[n + vll (m
(2), [>]![>]! ' ' * * WDj
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 13
Part II.
1.
The series
, = A{ ^ ' ] *> 3 L [2 ] [8,1] J X** + } * • (1)
has a general term
w iy, ,,+■ [""] [?tvl] [rev2r+l] J_ [ "" 2 ''
[2]W .... [2r].[2»l][2n3] [2n2r + l] A*f
which may be written if n — i> be positive and integral
/_ irA . r ^ [«v]![»]!(2) M [2»2r]! l a . D ^ fl
^ ' P [2»]! [n  r]! [n  v  2r]! [r]! (2) f (2)„_, X*
so that if we give to the arbitrary constant A the value
X"
[2n]l
(2)
[n]\[n v]l(2) n
we have the series
y — "V / _ 1 \r n r.r+2 I™. ILL . \n»2r~lnr»l
V h )P [nr]\[nv2r]\[r]\(2) r (2) n _ r A x
which will be denoted by P" («,X)
When n and v are not integral
[ "i ( ' n<r»])n([»  v]).(2)„ 1 * [2][2«i] x 2 " + ' • /
When p = 1 this function reduces to the function denoted m{nv) by Todhunter
{Functions of Laplace, Lame, and Bessel, p. 80).
We see that
^' P C > X > = X'O"*) .... (3)
Also
P; (asX) = A.' 5' .. ! P M («F.X ) ) . . . (4)
P;^k) = x^%^) .... (5)
For brevity P^ will often be used to denote V (x.\).
The function P^ satisfies the differential equation
d [2) Y i d m V dV
or
VV 2 ' 1 d(z» 2 Y 2) dW> I ' w
14 THE REV. F. H. JACKSON ON
When p=l the right side of the differential equation vanishes and the whole equa
tion reduces to
(x  \»)f + 2(v + 1)'^  (n  v)(n + v + \)y =
ax ax
2.
In this article we shall show that
1 _ Jn v][nv 1] ■> ,•■ [»  v] [»  v  1] [w  v  2] [»  v  3] iv _ (7)
P [2]t2«l] P +P ~ [2][4][2nl][2n3] P K)
of which the general term is
+1 [ra  y] [n  v  1 [n  y  2r + 1 . ,„
P [2][4J • • • [2r].[2nl] [SnSr+lf
(8)
(9)
may be expressed as a convergent infinite product
U([n])U([n + v]).(2) n
H([v])n([2n]) • (2),
reducing when n and v are integers to
Ml[nfcy]r.(2).
[v]I[2n]I(2),
as was anticipated in art. (10), Part I., by analogy with the sum of
, n — v • n — v — 1 n — v • n — v — \ • n — v — 2 • n — v3
" 2 • 2ra  1 ~ 2 • 4 • 2ra  1 • 2n^3
_ n\ n + v\2 n '
v\2n\
In the Proceedings of the London Mathematical Society, series 2, vol. L, " Series
connected with the Enumeration of Partitions," the following theorem is obtained : —
r=oo
[P*+v z ] m = [P x z]"' + 2 p "{ ,'" } [P^] m  r (P y )V . . . (10)
[ft] denoting 1 ^ (P ^ 2 + 1 A )(P ^ + 1) ' . . (P^ z + 1)
P>1
(j»jy n (P"l)(P»+'l) . . . (P^+^l)
o
P" 1 ' _ I . P'"' ! _ 1 pmr+ll _ 1
P'l • P 2 'l P r '1
PI I, l , ', I, h
» PiptPiPi ■ ■ ■ ' Pi
x » Pl X P2 r s
l x l I,
p x p . . . . p s
x 1 x a . . . . x,
ViVi ... ■ ft
being sets of s independent elements.
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 15
When m is integral and positive 4 [P*2] m reduces to the product of m factors
(P x z+l)(F x+t z+l) (P*+'^z+l)
Dividing both sides of (10) by [PV] m we have
[T>x+y„\m «•=«= pmi _ 1 p«r+l! _ J P y  1 pu+^ll
[P*z] m ~ X "*" <£ P  1 P"  1 P*+<"» 2 + 1 . . . . p^+mr+IJg + 1 V I
putting
Z = 2
to =
2
a = 1
y = n+v + 1
a; = » + v + 1
the general term of the series becomes
■pnr 1 pn— y— Sr+J _ 1 pn+Ml _ 1 pn+i>+2rl _ 1
( iypr{n+v+l) r — l r i . t 
V x / p2 _ l p2r _ J 1 _ p'.'nl 1 _ p2n«2r+l
/ nT rUr [»v][»vl] [«v2r+l]
1 ' [2] [4] . . . [2r] • [2» 1] . . . . [2n2, + l] r
which is the r+ 1 th term in, Const, x Pr n ,(l • p*~").
The infinite product L r ^ , when we make the same substitutions as in the series,
nPOOTUGS
L (P" +y l)(P" +1 ' 2 l) . . . (P"+»ac+"  l).(P"+»i 1) . . . (p»+a»+i_l)
r. (P 2 '  1)(P 2 ' 2  1) . . . (P*^ fc+ *lj(P* 1 l) . . . (P'"^+ 1 l) " 2 >
P>1
which since
p2* _ 1 . pSy— 2 _ l p2»— 2it+2 _ 1
= (P"1)(P" 1 1) . . . (P" K + 1 1).(P , '+1) . . . (P" K+, + l)
and
p2nl _ i p p2n— 2 _ I p2n— 2k+1 _ 1
_ (P 2 "1)(P 2 " 1 1) .... (psac+i_l)
= (P 2 "l)(P" 2 l) . . . (P"^+_l)
we are justified in writing
n([» + y])n(M).(2) n a3 v
n([2n])n([v])(2), • • ' •
by analogy with Gauss's II function.
Moreover, when n and v are positive and integral, the infinite product reduces to
[> + v]![>]!(2)„
[■2n]l [ v ]\ (2),
We have now
n([« + vj)n([n])(2)„ 1 _ Jn v ][nv\] 2 (U)
n([2»])n([v])(2). ■ p [2][2»i] p +  ■ K '
16 THE REV. F. H. JACKSON ON
and since
rw(3!A) " n([n])II([«#)„r y . [2][2*l] A 2 + /
putting
A = p*"
x= 1
we have
If v = this gives
P w (l J>*) = P> .... (16)
which was obtained in art. 9, Part I., by another method.
p ,» m _ n([n + ^]) (17>
When v=±
which when p = 1 gives
^ n"(»j)
1 .I^+i) .... (18)
T« — ; — TT/„, 1 \ x '
3.
Consider
rars^iisr * [B toix + 1 ■ ,19>
the general term of the series within the bracket being
I : r]l (2)„(2)„_
cW 1
Then if we perform the operations indicated by ■ v ) we obtain a series of which the
general term is
^ '[n]r [nr]l[r]l [»»2»«]l (2) 7 .(2)„_ r V ;
This reduces to
( ^ [ W _,.] ! [ w _ v _2r]![/]!(2),.(2) )( _, A
viz., the general term of
A" + "P [ "„ ] (yP" + ". A)
.?;„(.,.)  ^d{^'  ,w^:; + ....} • (22)
reducing when p = 1 to
2"A"P„(aA) = 1 D"+"  A./ 1 i" . . . (23)
The expression
72)7  pW (5KT • m
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 17
cannot be expressed as a product of n factors analogous to (X 2 x 2 — l) n , but it possesses
the property
^ ^Wt^^t + + (1)V'" +2  ! '
(2)„ r L J (2)„_ 1 (2) 1 [r]l[nr]\ (2) n _ r {2) r
_(A 2 p s )(\ 2 p r >) .... (A. 2 p 2 " +1 )
1
A 2
for z
3
,, 1C
2
„ I
w
a positive
integer for
m
2 >3X2. J p5_X2
„2n+l _ \ 2 1
P A / _ 1 \n x
(25)
(2)„
which is a particular case of
*[P, x a]»=l + 2 : P re+rj ^ I,z {"K • • • (26)
!' = 1
P2<1
substituting
r P 3_ 1_ " l + y ( ny.^ M[2"2] ■ • • [2n2r + 2] 1
L * A 2 J " + <f ( " [2] [4J . . . [2r] W
which may be differently expressed
(x'yXA'p") . . . . (x 2 ^»^) _ *> x»>
(2). " (2) B n "W). * '
If we use the symbol I (n+ " ) as the reverse of D (n+y) we have from (22)
I ( " + V (xP n+ " • X)  (^.P'X* 2 * 5 ) (A 2 jr"+i) ,
1 J.J* A) A M ! (2),, • • (28 >
jj 2  1 • p 4  1 . .
of which particular cases are
r.y  P * =P~and J PL** p)=p rirrSwoT • 29 >
1=1 [ ] * =1 [ ] _»J! L' M J' (2)n.(2)»
4.
It has been shown in art. 9, Part L, that if
* I A 2 [2][2n + 3] J
and/(x) denote
( A 2 2?i + o ;
then
p ^ ~ hz& + \ i[»4[»vi]}« + [»v] [_»,%/(*) ^>
If we change A to \p" this differential equation is identical in form with the differential
equation satisfied by P [n] (cc,\).
* Proc. Lond. Math. Soc, " Series connected with the Enumeration of Partitions," series 2, vol. i.
TRANS. EOY. SOC. EDIN., VOL. XLI. PART I. (NO. 1). 3
18 THE REV. F. H. JACKSON ON
The general term in the series Qf„(#.A) is
[, + i'r]![» + r]!(2)„ + 3 ,._,(_„_,,_„
[r]![2»+2r+l]!(2) r J
so that
Z* [r]![2n. + 2r+l]!(2) r * A **
Now the general term of the series (30):may be written
(31)
,,. [2n + 1 1! [n + v + 2r]l [n + r]\ (2)„ +r >y[ _„_ y _. !r _ 1]x _ ar ^
i [« + v]![r]![2n+2r + l]!(2) r ' V '
so that if we give to the arbitrary constant A the value
A „,_ 1 [n + v]!
[2»+l]I
and denote the series by Q (x'A)
then
Stort®^  Z, [2n + 2r + l]l[r]l(2) r j
and
^ = A'/^Q^A,/) ■ • • (34)
If we had chosen Q" [n] (x\) to denote the series which satisfies
v»l]
(33)
instead of that which satisfies
^F>y 1 d^y
P^^l  r, jJI + etc 
M
<fo (2) A 2 ^ (2)
equation (34) would have taken the form
— «egv i = A.^ 2 Q (/I] (^A)
Qim^j^) satisfies
= vj> 3 { qKV • V)  qEV • v 2 ) } ■ • • (35)
To find the sum of the coefficients of x in the series Q^, we make the following
substitutions,
m=  "J^±l
1 = 2
y = n + v + 2
z= 1
re =  n + v
in the series (11).
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 19
The general term
becomes
prln+K)
pmZ _ ^ pmr+li _  P" — 1 pv+rll _ J
p rx ! '. i ; — ' — — — ! — '. — ! : : z r
P*  1 P''  1 pi+mlZg __ Y m px+mrl z + J
■pnv1 _ Y pnvtr+l _ } pn+"+2 _ J pn+v+2r _ ^
P 2  1 P 2r  1 1  P'"
1 _ p2«— 2r— 1> /
p2 , [n + v+l][n + v + 2] [?i + v + 2r] ....
[2] [4] . . . [2r].[2» + 3] [2n + 2f+l] " ' v '
and the infinite product
[Pfe]™
becomes
L [v n 1] [vre 2] [vre2/c]
(37;
«— [2v] [2v  2] [2v  4] .... [2v2 5 .+ 2].[2n3] . . . [2n2«l]
so that we have this product
a , + 2 [rc + y+l]re + y+2] 4 [re + v + 1] [« + v + 2] [w + v + 3] [to + v + 4]
P " [2][2n + 3] iJ " [2][4][2« + 3][2n + 5]
n([2» + l]) "^Q^lj,*) (38)
n([>+v])n([W]).(2)/
5.
Now take the differential equation
= /(x)/(^ 2 ) (39)
and find a solution in the form of a series proceeding according to ascending powers
of x.
Assume y = A t a; Cmi] + &.&?** + . . (40)
Then performing the operations indicated on the left side of the differential equation
(39) we obtain from a term Ax [m] the expression
p[m] [m  l]Aaf 2[ "' 1+[  ]  ^[to] [m  l]A^'" ,] + {l[ n y][nv \]}[m]Ax Am ~ 1]+1
+ [nv][nvl]Ax™ (41) ■
which is
{p[m\ [m  1] + [to]  [m] [n  v]  [m] [_nvI] + [nv][»v 1] } A.c"" ]  i[»»] [m  1] Aa***"«
which reduces further to
{[ m ] _ [„ _ v]}{[m] [ n v l]}AaP«  r^O] [to,  l]Atf° S[m  2] . (42)
20 THE KEV. F. H. JACKSON ON
So that from the whole series we obtain
+ 1 [mj  [«  ■'] } (Kl  [  »  f 1]}A 2 ^M  J^K] [m 2  ljA^" 1  2 ' . (43)
Choose ?>tj = m 2  2
/«., = ;/; 3  2
Also choose [mj [m^ — 1] = so as to get rid of the term
tJtM K " 1]A^ [ '"' 21
yr"A
Then, in order that the expression may be of the form f(x) —f(x p ' 2 ), the coefficients
Aj A 2 A 3 . . . . eto. must be chosen so as to satisfy the relation
^..K+d [%il]A f+I = {[m,][nv]}{[m r ][nvl]}A r
p~ A"
since [ m d["'i _1 ] = ° either m 1 =
or ibj = 1
for the value »h = °
and
a _ _/" 3 x y^ D*  v  2r + 2] [n + v + 2r  1] ,
A " +I  jp=* X * [2,] [2,1] K
 A 1 1   [ n  y ][" + v+1 ] T P] + I (44)
[2][1]
/W " ^ r " 1 p'^ 1 ' [1][2] + i { '/
the general term of the series y being
A P r  r X r [nv2r + 2] [nv] • [n + v+ 1J [ » + r + 2r  1] ^ (4g)
"When n is a positive integer and n — v is even, the series (44) is C. T v [n] (x\) as is
evident if we consider that there are   terms in the series, and so by substituting
n — v
— r for r we reverse the series. The general term becomes
, ,(^)L,„_„ + ,, + ,. v ,,,, [2r + 2][2r + 4] . . . [„v].[n + v+l][n + v + 3] . . . [ 2n  2r  1] [m . y _ 2r]
W [l]p]...[»v2r> (4 °
^.^^^ [ K  y ]^_ v . 2 ]_. [2]^ ,, r+2x „_ y _,
\nvl][nvS] [i^
[2»2r]!
„[nK2r]
[„_,.]. [„v27]![r]!(2)„_ P (2);
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 21
which is the general term of
„ r .„,2 [n v ][nv2] .... [21
A * * ?  y ij Lwry 3]... l[i]  p ;/^) • ( 4§ )
If we had used the value m 1 = 1
m r+l = 2r+l
we should have obtained a series satisfying the differential equation (39).
The series being
. ( m A 2 \nv 11 rw + v + 21 m , ) ....
y = A^ x™— L ^LL l x m +  . . (49)
the general term being
/ n >y~ r A ["v2»+l] [»vl]r» + v+2l \n + v + 2r]
{i.) ,(„,) f2?+ll! x
(50)
If n — v be an odd integer and the series is finite, the number of terms will be
nv+l
2
For r substitute — „ r
The general term then becomes
A [«vl][ w v3] . . . . [2] xn . y . 2r [2»2»]! xin _ v . m .
Ap" " +3 4" >+1 [ B + v] [n + v  2] [1 [n  r]! [n  v  2r] ! [r]I (2)„_ r (2Y_
showing that the series is
const, x P* ix\) .... (52)
6.
In this and the following articles we shall give examples of the expansion of various
functions in infinite series of the generalised Bessel's Functions. The three expansions
to be considered are analogous to the following theorems in ordinary Bessel's Functions,
J.(«) = 2^Tirr T o( :C ) + 2" +1 ' n + l ! J i (•'■"> + 2»+ 2 ' • n + 2 \ J ^ x) + ( 53 >
S m J (a;) = 2"'} { J m (,) + jJ m+ ^x)+ m 2 ~J m+i (x)+ } . . (54)
2^7^"! = J «(^) + '2 J «+i( a ') + 2^T2! J "+^ a; )" t " • * ( 55 )
the symbol S m denoting m successive integrations in which no arbitrary constants
are introduced (Todhunter's Functions of Laplace, Bessel, and Legendre, art's.
418422).
When [*]» denotes !£=* . ? ^i p^l
p'  1 p — 1 jr — 1
22 THE REV. F. H. JACKSON ON
we have an identity
[• + //], = M. + Z^'" ^ ,y 7 ,,» i '"£1 M " M ' (56)
Subject to a proper interpretation of [x] m , this expansion holds for all values of m
provided the series be convergent, the condition for which is p l >\. If m be integral
M_ m denotes = — =— ; . . (57)
The theorem in its generality is discussed in a paper on " Series connected with the
Enumeration of Partitions," series 2, vol. i., Proc. Lond. Math. Soc. In the following
work we require only the simple cases in which m is an integer positive or negative.
For y substituting  2n
x 2n + 1r
I 2
to r (a positive integer)
we have
O + 2r 2n\. = [9n + 2r] r + ^y(»+ i 2r ] [ 2r ~ 2 ] • • • • • ^' 2 ''~^ + ^ x [2« + 2r] r _ s [  2n],
Now [2n + 2r] r _, is
[2»+2r][2» + 2r2] .... [2n + 2a + 2] = [n + r][n + r 1] .... [n + s+1] ■ S 511
and [2»] a = [2»][2ji2] [2re2s + 2]
= (l)^'" N ' 2+s [2w][2?i + 2] . . . [2rc + 2s2]
= (_l)*p**^H[»][„ + l] . . . [„ + ,l].(g"±t>
So that we have
(2),[r]! = [ n + r][n + rl] . . . [?i+l f 2 ^+ r
{ * In
+ 2/ ^ [r.]I[.]l (2) r _ s (2) s LM + ,J " • • • L " + ' S+1J (2) n+s
[»][» + 1] . .. [w + t sl]( 2 ^±£
V.^/n1
Dividing throughout by [n +r]! (2) n+r [r]! (2),. this reduces
1 1 ' , l
[n + r]l(2) n+r ~ [n]l(2) n [r]\(2) r + 2< " ^^^[r _ s]! ^j, ( 2 ) r _,( 2 ).[» + *]! (2)„ +s *
M[ w+ i], [w+g i3 (2Wi (58)
V^/nl
Dividing throughout again by [r]! (2) r this identity becomes
1 = _i .  +
[n + r]J [r]l (2) re+r (2) r [»]! (2)„ [r]! [r]! (2) r (2) r
6f " [ W + S ]![ S ]!(2) K+ ,,(2) S  (2)^ W " + *" 1
• (59)
[r*]![r]!(2) r _,(2)
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 23
From which we can see at once that
T / W W* T / »" \\ 2 A»+V n+11 r n (2) n r , _H+1 , x
+ . ■ (60)
the general term being
subject to the convergence of the series ; for if out of each term of the series on the
right side of (60) we pick out the part involving \ n+2r x [n+2r \ we get a set of r+ 1 terms
Xn+2r„[n+2r] f 1 * _ „2 1 U1 ( 2 )» 1
1 [»] (2)„ * [r]l [r]! (2) r (2) r * [» + 1]! (2)„ +1 L J (.2) 1 (2) H+1 [r  1]! [r]l (2) r _ x (2) r
+ } • • • • (61)
which by (59) reduces to
*" + ^ C " +2]  . . (62)
the general term of J [n] (x\), — and theorem (60) is established.
A particular case of this is
*■**> = wfc^*flmw M <H>T'<''" !A > + (63>
analogous to
J B (s) = j^ Jo(«) + 2^ 6 Jl(a:) + * (64)
To investigate a theorem analogous to
S m J (x) = 2'"  j m ( x ) + m j m+2 (x)+ m ™±±J m+i (, )+ . . . . } . (65)
If, in the series on the right, we replace the Bessel's Functions by infinite series and
collect the terms together according to powers of x, we find that the terms involving
x m+2r f orm an i n fi n ite series
2V+ 2 ' i     + M + 1 I I (66)
(m + r\r\ 2"'+ 2 '' 1! m + r + 1! r 1! 2 m+2r 2! ra + r+2! r 2! 2 m+2r J
which is
2 g ( 1 _ m r (. / 67 v
2 m+2r r! r!ffi + f.ffl + r l.f+l 1 m + r + 1 . . . . . . r + 1 1
The series within the bracket is by an extension of the notation of Vandermonde's
theorem (Proc. Lond. Math. Soc., vol. xxvi. p. 285),
(»")«  ijWmlH + 2! ^>"^ r) l + (6 ^
which is (r + r)_ m subject to convergence conditions, viz., 2r + l >0
(r + /•)»> = (2r)_ m = — — —  — ^— y
m + 2r ■ m + 2r  1 . . . . 2r + 1
24 THE REV. F. H. JACKSON ON
So the rearrangement gives us an infinite series, of which the general term is
x m+ ' 2r 1
2 2r r!r! X m + 2r  m + 2r I 2r + l
which is the general term of S"\J (.r) ..... ( 6 9)
S" 1 indicating vi successive integrations.
7.
The preceding analysis shows us how to construct the analogous series for the
generalised Bessel's Functions. Consider the generalised form of Vandermonde's
theorem
5=00 ml 1 m— It i m—s\ll 1
[«+*]. = M. + 2><"» • v J i J {"** i M™M • (70)
s=l f L ' 1 J L • • • Jf 1
convergent for all values of m if p l >\ :
If m be a negative integer,
^~ m = [x + ml][x + m^ll] . . . . [x + l] ' (/ '
In the theorem (70) replace m by —ma, negative integer, I by 2, x by 2r, and y by 2?\
Then
5=oo
2m _ 1 r) 2m2s+2 _ 1
s(2r+2m+2s)JP 1 ■ • i 7 *
^ 2  1 jr'l
x Q]>2] [2r2.+ 2] ,
[2r + 2m + 2s] .... [2r + 2] v ;
Now [2r + 2r]_ TO is [4r]_ m
1
[4r + 2m][4? + 2m2] .... [4r + 2]
and this may be written
(2)gr_
(2) 2 , +J „[2rfm].[2r + ml] . . . [2r+l]
Dividing (72) throughout by [r]! [r]! (2), (2) r we obtain
1 1
[r]! [r]! (2) r (2) r X [4r + 2m] [4r + 2] =
1: + V(l)y +s+2r [?n+gl]!(2) m _ 5 _ 1 ,„,
[r + ro]! [r]! (2) r+m (2) r Z. v ' e [ m  1]! (2)_ 1 [»]1 [>«]! (2) s (2) r _ s [r + m + S ]! (2) r+m+s ^*'
the analogue of
1 1 = 2 m I 1 _ ™ 1 + I (7A\
2 2r • r! r! m + 2r . . . . r+l I m + r\ r! 2 m+2r 1! to + ?+ 1! r 1! 2 m+2 '' ' ' " ' j ^ '
used in art. 6 in the analysis of the series (65).
At this point we define the function J^xX) as the convergent infinite series
r=ao g "+''l
T * w+ ' r s ' 1 .... (75)
^[» + r]![r]!(2) r (2) n+ " r
GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 25
Then
r=ao [2ra+4r]
jfM) = 2 A " +2 '^ [2] ■ • • (76)
J[ " A ^[n + r]![r]!(2),.(2), !+1 .
j£0*) " P 2  m M • r.y^v Jfi(*Ap) + • (77)
Consider the series
I 2 )
(2) m _ 1 (2) 1
of which the general term is
( ^ [,] (2) m _ 1 (2;. J ^M^P) • (78)
If we replace in (77) Jf m] Jf m+2 ] by the infinite series which they represent
[2ro+4r ]
and collect the terms according to powers of x, then the terms containing x [2] form
the infinite series
[2m+4r ] s= oo [2»i+4r]
X'»+ r ./ ; ~ir + V ( _ iv A m + 2 ^ m ■ [m + s  1]! (2) m+s _ t p*+H*. ( 79 )
[m + r]! [r]!(2) m+ ,(2) r £* [m  1]! (2) m _ 1 [s]! (2)J>«]1 (2) r l,[m + r + «]l (2) m+ , +s
which reduces by (73) to
[2m+4r]
1 X"'+ 2r r~P]
[2m + 4r] [2m + 4r  2] • • • [4r + 2] [r]l [r]! (2),(2),.
which is
^pTJgS(a^".X) • • (80)
I m denoting the operation reversing
d(xP 2m ') I dia****) I ' " " ' I *W) ) J i
8.
It is well known that
2". re! 2 " + 2 2  2! 2 3 • 3!
Consider the series
J[»M) + Sh*^) + ^/^.•W***) + (82)
of which the general term is
^ (2) \rV? ln+r] ( x> ' ^
J [n] (ccX) denoting
V y [re + r]![r]!(2),(2)„
(83)
(84)
Replacing J [n] (x\) J [n+ i](x p X) ... in (82) by infinite series, and collecting the terms accord
ing to powers of x, we have for the terms involving x [n+2 '' ] a group of r+ 1 terms, viz.,
^n+2r a Jn+2r] ^n+2r x [n+2r]
( " l) \n + r}\ [r]! (2) r (2) B+r + ( " lr \n + r]![r  lj!(2) P _ 1 (2) )l+r (2) 1
+ ... + /_ iyv'i A * (85)
+ " + ( l ) P [ ra + r]![rs]!(2),._ s (2)„ +r (2),. V '
TRANS. ROY. SOC. EDIN., VOL. XLI. PART T. (NO. I). 4
26 THE REV. F. H. JACKSON ON
which is
which is identically zero.
The only term which does not vanish is the first term in J [H] (a;X) so that we have
identically for all values of n
*vM) + ^St. W**) + (87)
[»]!(2)„ l " lv ' (2) x [l]!
If in 84 we had taken J [n] (xX) as
'[re + r]![rj!(2) r (2), H
the signs in the terms of (86) would not have been alternately + and — ; (86) would not
then be zero but
2(» 2 +l) . • . Or r ~'"+ 1)  
Another expansion x [n] in terms of the P functions is
[2n]!X»a™ _ ,,., 3 [2re3] +rfl [2nl][2n7] p (8 g,
analogous to
2re!z* 1 D 2re3 D 2?i  1 • 2?J  7^ ,
= "i. + — "n2 + H 5 "«4 +
re! re! 2" 2  2 • 4
9.
Various interesting theorems have been obtained with respect to Bessel's Functions
j_
when the variable is not x but *Jx. The analogous theorems for J [n] (xp+ 1 ' A*) are given
in the following work. It is well known that
iJx ,„\x ^(^)) = (\) m x— J„ +m (» . . (89)
,7m f " 1 n— m _
£n\ X *UJ*)\ = (ir*~ J»»( */*) • • • (90)
Let D« denote ^^ . ^ .... ^  ^ .... (91)
Then
which reduces to (89) w T hen p = 1
B »] 1_
V A a; ^^"iTg "' 1 . . . (93)
« [r]l[n+r]I(2W2W
GENERALISED FUNCTIONS OF LEGENDPE AND BESSEL. 27
[2r]
_, \ r xP n [2]
' ^[r]![» + r]!(2)«(2) B+r ' (94)
Operating on this with D tml we see that the operation reduces to zero all terms for
which r<jn, while the result of operating on the general term
[2to+2t ]
[m + r]\[n + m + r]\(2) m+r (2) n+m+r ' ' * V°>
is
\2m + 2r\ \2m + 2r  21 \2m + 2r  2m + 21
rTTn " • tTvt " • • • • L F7 r=. [ 2m+2r2m]
V* [ 2 1 [2] [2] ( X p' l+2m ) —m , (96)
[m + r]! [n + m + »•]! (2),.(2) n+m+r
which is easily reduced to
[2]" A * ^ [r]![ W + m + r]!(2) r (2) B+m+r ' , ' ' "">
viz., the general term of
' ^;rj [rt+ i^W .... os)
. [ 2 ]
In a similar way theorem (90) may be generalised
We have shown that
[»] • P W (*A) = k[2n  ljaiP,,,.^, A.) p" + \n  l]P [n _ 2] (*A) . . (a)
In the same way we can establish
i^MQwO"*) = X>  lJcQ^^X)  [ n  lJOu.flfoX) . . (0)
analogous to the ordinary recurrence formulae.
Multiplying (a) by Q [n . 1] (x p ,'X) and (/3) by P [ , l _i ] (x^,X), then subtracting, we obtain
[n] { P w (a,X)Q [ „_ 1] ( !B ",\)  y^^P^^X) }
= [«l]{P [n _ 1] (^A)Q [w _ 2] («X)^+ 1 Q [ „_ 1] (^,\)P [ „_ 2] ( a; A)} . . (y)
analogous to
»{P«Qnl  Q,,P„,} = (»  0{PnlQn2 " QlP.«}
So also multiplying (a) by Q [(l _ 2 ](as,A) and (/3) by jt) n+1 P [w _2]( a; >A)» * nen subtracting, we
obtain
[n]{P w (a« l X)Q Di _^sB I X) p 2n+3 Q m (x\)? ln v(xk)} = A[2n  l].r{P [/i . 1) (.r"A)Q [ „_, ! ( a; .X)  p n+i q in  n (x^k)F [n ^(x,X)}
_w ij
When we put as = 1 we obtain
[n]{P M (X)Q tn _ 1] (A)  p" +2 QMVm»W} = [«  1] {P[»n(X)Q [n  2] (X)  P n+ % n i^Wm^)} («)
28 GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL.
and so by repetition, if n be integral,
= Li]{P w Qto]i> 8 Q w iW
= 1
the analogue of 1',,,+uQ.  Q»+il J „ = n + \
Similarly from (S), putting x = 1 and repeating we obtain
P [ , 1+1J (A)Q 1 , l  1 (V) j^Qj^XJPp^X) = H s " + fl . (,)
Ln+lJL»J
From the recurrence formulae we can obtain without difficulty
"[2r+l] [n + 1] P [ , t+1 ,(A)P l „ l (X')iy, +1 (A')P M (A)
A theorem analogous to Neumann's expansion of an arbitrary function in a series of
Bessel Functions I have given in a paper, L.M.S. Proceedings; also the analogue of
Lommel's theorem
^ = {J^«)}" + S{J f (a)}* + 5{J f («)}»+ ad inf.
( 29 )
II. — Certain Fundamental Power Series and their Differential Equations. By the
Rev. F. H. Jackson, H.M.S. " Irresistible." Communicated by Dr W. Peddie.
(MS. received December 7, 1903. Read January 4, 1904. Issued separately February 16, 1904.)
The series which will be discussed in this paper are of the general type
Aj**"") + A 2 x l "^ } + +A r x <mr) + . . (1)
Consider a sequence of elements Pi,p 2 ^P3, then (w r ) will denote the sum
of the first m r elements of the sequence. The simplest connection between the terms
of the sequence is equality
Pi=P2=lh= =Pk= • • •
The series expansions of the ordinary functions of analysis and the series which are
solutions of ordinary linear differential equations belong to this simplest type. Another
more general type of series is formed when the elements pj,p 2 ,Ps • • • • are m
geometrical progression as a, ap, ap 2 , In this case the index (ft) denotes
a + ap + ap 2 + + ap n ~ 1 , which is a r , the limitation of n to positive
integral values (where the sequence is geometrical) may be removed, as will be seen in
the particular discussion of the functions J [n] (\x), P [n] (\x), Q [n] (^x), F([a] [/3] [yjAa:). lip
be made unity, the geometrical progression becomes a progression of equal elements
and the properties of the general functions reduce to analogous properties of the simpler
functions ~F(aftyx) ; P; t (x) ; Q n (x), etc. Euler's expansion
(1  x)(l  a; 2 )(l  x s ) ad inf. = ^(  l)".rf' 3 " 2+ ' !)
(x < 1)
and Gauss's series
( la;' 2 )(l a: 4 )(l x 6 ) . . . ad inf. _ y a .'^i 1
(1  x)(l  x 3 )(l  x?) . . . ad inf. ~ ^
(x<\)
are particular cases of the general series (l) for the sequences 1, 1, 3, 2, 5 . . . ,
2n — 1, n, . . . and 1,2,3,4, . . . . respectively.
The fundamental Hypergeometric Series is
F 1 + a l' a 2' a 3 % ' 1 >+ a i( a i + (1) ) a,(a,+ (1)) (2)+ ( a\
MPiPfPs p t (P(2) y 8 1 (^ 1 + (2)(l)) .... £(/J 5 +(2)(l)) ' VW
In the case when there are only three elements a /3 j in the Hypergeometric Series and
the sequence of p.s is 1, 3, 5, . . . . , 2ft +1, . . . the following series are interesting
cases :
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2). 5
30 THE REV. F. H. JACKSON ON CERTAIN
U «i, + aa + 1 pp + 1 . aa+1 a + (n  l) 2 • /?•/? + 1 /? + (rcl) 2 , l2 m
l 2 y l.2.y.y+3 ' ^•••■■ r  l.^.3 ;/yy + 3 . . . . yl+W '"' V/
+ l 2 y + l l 2 2 2 .y+l 2 y + 2 2 W
l + a ^ + a . a+ l ! .y + l V + (5)
It is well known that if II g (a + m) denote the product of s factors
(aj + m)(a. 2 + m)(a s + ra) .... (a„. + to) to integral
an identity of the following kind can be established,
II 5 (a + to) = B + B x to + B 2 to(to  1)+ .... +B 5 to(to1) . . . (tos+1)
where B Bj . . . . B 4 are constants, that is, are independent of m.
We proceed to establish a generalisation of this, on which all subsequent work will
depend.
Let n„(a + (m)) denote the product of .s factors
( a l+Pl+P2+ • • • • +Pm)( a 2 + Pl +P2 + ■ ■ • +Pm) (a s + p 1 +p.,+ . . . +V m )
then n s (a + (ra)) is identically equal to
B u + B (OT > + B (m)((TO)  a)) + . + b (ot)((w) " (1)) ((■>») (*!)) _ (6)
1 Ih P\Pl P1P2P3 ■ ■ ■ Ps
which may be more conveniently written
B + B<^ + B 2 ^ + + B (™h . . . (7)
Pi P1P2 Ps*
The coefficients B B 1? etc. are independent of ra and are given by
gB„=< 1)V ^n(, +(> ,,,). . . . (8)
Before proceeding to obtain these coefficients it will be well to explain the notation
clearly.
(m — 1) is not the same as (m) — (1), for (ra — 1) denotes P1+.P2+P.3 + . . . +p m i ,
while ( ra)  ( 1 ) denotes p. 2 + p 3 + p 4 + . . . +p m
(m), = (p 1 +i? 2 + . . . +p m )(p 2 +P s + • • +p m )(p 3 + ■ ■ • +P m ) • • • • (p,+ ■ ■ ■ • +Pm)
B s ^ ! n(a + (3)) ( ^ r! n(a +( 2,) + (J J^ J] n(a + (i) ) Mn W . . (9)
(3)! = 0?! +^2+^8)^2 +^3)^8 I 3 }' = (P S +P2+lh)(P2+Pl)Pl
(2)! = ( Pl +p 2 )p 2 {2}! = fa+pjfa
(1)!=P, {1)1 JP. .... (10)
In the expression for B 4 * four elements p 1 , y? 2 , p 3 , p± appear and
(4)! = (p 1 +ia 2 +p % +p i )(p 2 +p z +pd(p 3 +PM I 4 }' = (Pi +P& +P2 +Pi)(Ps +P2 +Pi)(P2 +Pi)Pi
(3)! = ( Pl +p 2 +p 3 )(p 2 +P 3 )p 3 {3} ! = (p A +p 3 +p 2 )(p 3 +p 2 )(p 2 )
(2)! = (p 1 +p 2 )p 2 {2}! = (p i +p 3 )p 3
(1)1 = Pi {1}!=^4 (H>
FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 31
The symbol { } ! must not be dissociated from the expression B in which it occurs,
because {r\ ! as formed from n elements p x p 2 . . . p n is not the same expression as
\r}\ formed from m elements p x p 2 . . . p m .
In general, for n elements
{1}! =Pn
{2}! = (Pn+Pn^Pn!
{3}! = (f»+j'»l+P»!)(ftl+ftj)j), l  2 . ■ • (12)
At this point some properties of the coefficients
_(«)!
(»
may be noted
(nr)\ {r}\
(n)l
which is the generalisation of
^ K '(nr)\{r}\
2(iy
»!
=
r\ n  r\
As an example of this property take (p 1 , p 2 , p 3 , p 4 ) = ( 1 , 7 , 9 , 5), then
The expression
(4)
(3)
(2)
(1)
{1}
{2}
{3}
H)
(4)!
= (l + 7 + 9 + 5)(7 + 9 + 5)(9 + 5)5 = 222M45
= (l + 7 + 9)(7 + 9)9 = 17169
= (1 + 7)7 = 87
= 1
= 5
= (5 + 9)9 = 149
= (5 + 9 + 7)(9 + 7)7 = 21167
= (5 + 9 + 7 + l)(9 + 7 + l)(7 + l)l = 221781
(4)!
(4)!
(4)!
(4)!
IS
(4)!{0! '" (3)!{1}! T (2)! {2}! " (1)! {3}! T (0)! {4}!
, 1078 55 55 735 n
1 — 4 — _ — + =
408 + 12 4 68
If p x p 2 , . . . be an arithmetical sequence 1, 1 +a, 1 + 2a, etc., then
Z (»)! (2K»l)a)(2 + «a) .... (2 + (2n2)a)
(nr)\{r}\ ~ (!+«)(! + 2a) .... (l + (Ml)a)
a generalisation of
•I n — r\
an.
(13)
(14)
to which the identity reduces when the sequence (P1P2P3 • • • ) = (1 > 1 > 1 •
If the sequence is (p x p 2 ■ ■ . ) = (1 , 3 , 5 , . . .)
'(r)\ {nr}\
= 2 2
•)•
(15)
32 THE REV. F. H. JACKSON ON CERTAIN
The identity (14) is a particular case of F(« /3 y 8 e 1) and may be thrown into the form
(x)„ = (2.r)„ + 2^^^ 2r)„_ r (2.r r+1), . . . (16>
iii which
(.«)„ IS
xxYxl . . . xn + l
A more general form is
(17)
(m
M. = DH. + z ^ f3l g J : r3 t fa  2 *W 2 *  '• + ^ •
r i _ 1 r  = P'I
" = M [* !] [»» + 1] [ft:J ~ p  1
These and other interesting theorems due to change of the sequence (p x p 2
must be left to another paper.
We now proceed to obtain the coefficients B B x , etc.
Suppose that II s (a + (m)) is capable of expansion in the form
Etf. + (»))» B + B 1 ^ ) + B 2 ^ + + gM. .
Pi i>2 ! A 1
(m) s denoting (^ + . . . +p m )(p 2 + • • • +Pm) . . . . (p s + ... + p m )
PJ „ Pi'PiPz • • • ■ Ps
In (18) substitute (m) = 0, then we have
B = n s (a)
Similarly, if we substitute (m) — (1) = we obtain
b +B i a=n(o+(D)
Pi
Continue the process of substitution by putting successively
(m)(2) =
( m )_(3) =
We obtain the following set of equations for determining the coefficients B B l5 etc.
n(a) = B
n(a + (l)) = B 0+ ^B 1
Pi
U(a + (2)) = B +Pl±l2 B + Pi +P2P2 B .... (19>
Pi P1P2
n(a + (n)) = B + ( tt )B 1 + ^B 2 + .... + @?B n
Pi Pj P» l
FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 33
From these equations we obtain
B = n(a)
B 1 = n(« + d))n(a)
B PiPi n(g + (2)) giggn(a +(!))+ PlPa n(a) . .' . (20)
Pi + P?P2 PlP2 Pl'lh +P\
_ ,j n(a + (3)) n(a + (2)) n(g+(l)) 11(a) I
3 P3 \ (3)!{0}! (2)!{1! + (1)!{2}1 (0)! {3}! J
To establish the form of the coefficient B„ we assume that the law of formation holds
for BoBjI^ . . . . B n . v namely, that
,/ n(q + (tti)) u( a + (n2)) , t n(g) i , n
HniPni\ („_!), {0 }! _ (n""2)TpT! ( ' (0)!{«l}!l ' ( >
. ,/ n(a+(»2)) n(a + ( «3)) 4.(i\n* n(a) i
" 2 ^ 2 '{ (re2)!{0}! (n3)!{l}l V ' (0)! {« 2}! J
From (19)
n(a+(K)) = B + B 1 ( !i ) + +B r ( 4'+ • • • • + B«£&
Pi ft! ft!
Replacing B B 1 . . . . B^^ by the expressions (21) we obtain
B„ = A.II(a+(ra)) + A. l II(a+(rcl))+ + A s II(a + (n  s)) + .... + A„II(a) . (22)
where
A 1= 
ft! (»)»!
1 (»)!(» 1)!{0}!
X _ P» ! ( OOnl _ (")«2 I
2 (n)ll(nS)I{l}! (rc2)!{0}!j
\ _/ _ ly+li 5 " / ( w )nl _ W»2 ■ i / _ I \s \ n )ns \ /oo\
5 v ; (»)n(n«)l{*l}~l (ns)\{s2}l K '(ns)[{0}lj ' { '
{s — 1}! in the first term of X s is formed out of the set of elements p^ p 2 . . . . p n \
and is
(ftl+ft2 + ■ ■ ■ + Pns+l)(Pn2 + ■ ■ ■ +Pns+l) Pns+l
{s — 2}! in the second term of A s is formed out of the set of elements p 1 p 2 .... p n z.
and is
(ft2 + • • • +Pns+l)\Pn3 + • • • + Pns+l) Pns+l •
We see that \ which is
ft! («)«!
(»)!(»l)!{0}!
may be written
ft! . ( Pl+ ■ ■ ■ +Pn)(P2+ • • • • +Pn) • • • (ftl+ft )
(nl)l ( Pl + . . . +p n ){ p . 2 + . . . +p n ) . . . (p n ^+p n )p n
ft! ft!
(n 1)1 ft (»l)l{l}l
since p n is {l}! for the n elements p x p 2 . . . . p n .
34 THE REV. F. H. JACKSON ON CERTAIN
Similarly X 2 may be written
PJ / (»)»! _ ( »)»g I
(w2)!l (»)!{1}I (»)!{0}!J
which is
/'« ! f__L_ _ —J— I
(m2)! j yv/)», Pn1+iV.ftJ
(*
J»n!
" (»2)liWi+lVJ>Wi (»2)!{2}!
since (_p„ _ x +p„)/?„ is {2}! for the n elements PiP 2 . . . . p n .
In the expression (23), if we reverse the order of the terms we have
and since
in K,
(n8)!l(n)!{0}! (n)l{l}! V ' (»)! {* 1}! ' " V
H.i _ M
Vyp = Pn(pn + Pni) (p» + P„l + + /'«»+l)
also
{0}! = 1
{1}! = £>„_ s+1 (Because the term involving {1}! was derived from B„_ s+1 )
{2}! = (/V s+ 2+iV s +iK_ s+ i
{3}! = (p n s+S+P n s+2+Pns+l)(Pn S +2+Pns+2)P n  S +l
{S1}\ (pnl + + Pns+l)(Pn2 +  ■ • + Pns+l) P«s+l
We see that for the set of elements p n p n . 1 Pni+i
the expression
\ n )>i s _ ( n )ns+l , , (_1\J1 \ n )nJ
(n)\{0}\ (n)!{l}I ■*"•••" "^ > (n)I{«l}l
is
W " (— 1)!{1}! + +(  ir \l)l{sl} ■ ■ • (25)
but for any set of s elements in any order
+ + (1) s t4 = °
so that
®  Fwri + + ( y*W=m   (  1> 'ft ! • ■ < 26 >
and we have
A. = (i)V
Vn\
(ns)\{s}\
which establishes the form of B rt in general
b,  (i>' (  i dyrw! n (" +( " s »> ■ < 27 >
FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 35
The General Hypergeometric Series.
Let (p denote the differential operator
B ft + B 1 aj'"D (I) + B 2 a*'+^D (2) + +B s a; ,s »D (s) . . . (28>
and vf/ denote the operator
^c x ,1, D 1) + C 1 a; ,  ,, D« 2 »+ + C^'D'" I (29)
B„ being' £( If ? ^r^i n ^ a + (n " r » ■ ■ ■ ( 30 >
r=0
° § (  1) V^W w+< * ,  r>) • • <»>
Then if y denote a series of the form 2Aa; (m)
that is
A 1 a; (m ' , + A 9 cc , ' %) +
and we operate on y with <p — \J/ we obtain
{^  f }y= T A.n s ( a f (m))x<»"  T A ( ^n^ + (m)  (1))^'H . . ( 32 )
The lowest index present in the series on the right is (w^ — (l), and the term in which
it occurs is
A 1 (iin,(£+ (wij)  (l))a ,m >> (1 >
Pi
Choose mj so as to make this term vanish, the possible values of m 1 are zero and
(1) — j8. Now, from the manner in which (m x ) (w 2 ) .... are formed, we cannot have
any such relation as (m r ) = (m r+1 ) — (l) unless the elements p 1 p 2 p 2 . . . . are equal to
one another, so we choose
(»» 2 )=< 1 ) = /'i
(m 3 ) = (2) =j>! +i> 2
and
A
Pi
A r+ ^r±^U t (/3 + (m r+1 ) (!)) = Air^q + (m r ))
Then we have
F = l + », n ^ g ) gin + » 2 n,(a)n,(q + (i)) (2) , 33>
1+Pl (i)iW +A (2)in < 08)n^ + (2)(i)) a ' + (dd>
the general term being
» n,(a)II,(q+(l))II,(q + (2)) n s (a+(nl)) r{n) ,
^ (»)!n,C8)n,( j 8 + (2)(D) n,G8 +(»)(!)) ' '
and a differential equation
36 THE REV. F. H. JACKSON ON CERTAIN
(«♦>'"*> { o ^ flffft?! : : : : : %? J ^ ■•■•)
(■ ^' + ( '&;r?::::::::r"' ) — •••)} ■ (35)
In the simple case
F = 1 + p£&& + ft 2^+#+?L^4 (36)
py PiPi+P27y+p2
the differential equation is
a(3Y + /y((a + (3 + 1 j 1 )x"^  y}D (1, F +p x 2h{ x ' h  1 }^ 2 D (a, F
= aflla + „ i <j±M±vy, + p i2 ^±a a+ Pi + ^/3+^^+j>i+p 2a!P ,to + . . .)
H \ y i\y Pilh+lhyy+lh
_(1 + p a ±M±Plr>', + « 2 « +ffi<* +ffi + JVff +i j i/? +Pi +V ?, + . . . .) I . (37)
Puttino x = 1 the expression on the right vanishes identically, therefore when
[D (2) F] a:=1 is finite or convergent we have
«/3[FU + pM + Py+Pu
which is
dx w _
x=\
= . . . (38)
i + « s ''+j ) i/ i '^j ) i + = Pi(y a PPi) {1 +p a ±PiP + Pi
PiPi+P2yy+p2 y ' x iYy + i>2
a/3
] Pi7
+ p 2« + ?V a +Pi +P 3 P +frP +Pi +P 2 + , .,(, . (39)
1 PiPi+Piy+Pay+Pi+P* ')
subject to the convergence of the series.
When the progression of elements Pip% is 1 3 5 7
]?<, a y+ i, ,■) = 1 4 %, + t ?; +1 /f + V + • ■ ■ •
l'y+l l z 2y + 1y + 4
F(„ft !,,, , 1 + tf, ♦ ' + I ;f±ix. + ±i^»±M±lV + (40)
In connection with this series we can obtain from (39) the identity
1 + {uf + (ST** + (sir  *""
s/l l. »l gl_,. »l 2 »2 3 xl 2 a;2 2 , »  Vn  2H  3 2 a> l 2 a 2 2 x 3 2 ^ )
 (j: + n; j + ^ ^ + ^^ ^ + 122 2. 3 2 2 2 3 2 4 2 ' ' J { '
putting n = — x = m 2
, //t 4 m 4 m 4  1 4 _ ^
a\y (2i)* "
In the case when Pjjt^ form a geometrical progression and F([a] [/3] [<y]Xa;)
denotes the series
1 + M [/3] A ,,n . M [« + *] [0] [0 + Qxym + ( ± 2 )
WW HPiHrKy+q ( }
u p 1
the following relations can be obtained :
FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 37
F([a][fl[ r >r— "<) = fr " ^L^^l^" ^ 1} f (M 08] fr  *1j»— '') • (43).
F([a]L8][y>v— ^) = ^Jr_^^L ] . F([a][ ^ ][y]py a 3 
F([a][/i][7>v — *) = *
F([«]C8][ 7 a]py/i)
and other similar relations found by interchanging a and /3.
The expression of F([u] [/3] [7]p y_<1 ^) and F([a] [/3] [y]p y ' a ~ p ~ l ) in the form of infinite
products analogous to ^\Zf™ is effected at once by the above relations. The
investigation of these I have given elsewhere, but note the results here as of interest in
connection with the Fundamental Hypergeometric Equation discussed in this paper.
Particular cases of these series are
r<[,]) = (jf My* m 2 my* . i !
T([x + $]) \[x] [l]![as][a:+l] [2]! [x] [x + 1] [x + 2]
l> 1]' if + [JlV 1 , Mf]'^ , l J (44 v
i m rii!Hk+n r2iiTiira!+nraj+2i ■ ' • ) " v r
Bessel's Series : —
Consider now a progression
P6 P5 Pi PA i'2 Vl Po Pi Pi lh
denote
P1+P2+ ■ ■ ■ +P„ by (ra)
and
P0+P1 + • • • • +P,,+, by (71)
lhen the operator
Apjjjja^D'"'' + \{(1) (»)( »)b 1 a ,1 'D (1 ' + X(n)(  ra)D u > = <£ . . (45>
operating on A^"" gives
A[(m 1 )((m 1 )  (1)) + {(1)  (n)  (  n)}(v h ) + («)(  ijflA^
which is
So that if we operate on a series y = ^ Ax (m)
$"J = 2t(™)  (»)] [(»»)  ( " ?i >] Aa:( ""
choose
(//« T ) = (?j) and (m. 2 ) = (ra + 2)
K) = ( M + 4 )
AA, +1 [(m, +1 )  («)] [(m r+1 )  (  »)] = A'
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2).
38 FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS,
then
( ~(»+2) I
J \ + [ (w + 2 ) (»)][(n + 2) (»)] f
and
( r (»+4) )
*» = iA { ^ + V +2)  W H(» + 2) (  »)] + ' ■ ' } ' ■ ■ (46)
when the elements are
Pi = Pi = P 3 = 1
Fo = P\ = P2 =  l
The series is Bessel's Series and <P'y = ^x 2 y is Bessel's Equation.
In the particular case of the progression of elements PiP 2 .... being
P3 P2 P\ PO Pi Pi PS • ■ • •
7531 1 3 5 . . . . andA= 1
I ~(n+2)s ,j.(«+4) 2 ' )
V = \ X (n + 2)*n* {(w + 2)*ra 4 }{(7i + 4) 4 n 4 } " J " ( '
( 39 )
III. — Magnetization and Resistance of Nickel Wire at High Temperatures,
By Professor 0. GK Knott, D.Sc.
(Read May 4, 1903. Given in for publication November 12, 1903. Issued separately March. 3, 1904.)
In a recent paper published in these Transactions # I gave an account of certain
experiments upon the change of electric resistance of nickel due to magnetization at
different temperatures up to 100° C In the closing sentence of that paper I pointed
out the advisability of trying to push the temperature up to 400° C, the temperature
at which nickel loses its pronounced magnetic properties. This has been accomplished
in the experiments now to be described. These refer, meanwhile, to the effect of
longitudinal magnetization on the resistance of the wire. The experiments on the
effect of transverse magnetization are still incomplete, and are reserved for a
future communication.
1. The Apparatus used. — Since the temperature was to be raised to about 400° C,
it was necessary to use asbestos for insulation. Accordingly, two exactly similar anchor
ring coils with nickel wire cores were constructed. These coils were about 18*3 cm. in
diameter. Eound the flat circular coil of nickel which formed the nucleus of the
anchorring two independent layers of copper wire, carefully insulated throughout, were
coiled, with the same number of turns in each. The ends of the two copper wire coils
could be joined in different ways, so that it was possible to have a strong current pass
ing through both, and yet to have, at will, either strong magnetization within or none
at all. The magnetizing force could thus be removed at a moment's notice by simply
reversing the current through one of the coils of copper wire, while the heating effect
of the current on the whole coil could be maintained unaltered. To preserve as con
stant a temperature as possible during any one set of observations was of the highest
importance, for the change of resistance due to a very slight change of temperature was
sufficient to mask completely the change due to magnetization. This change was
measured by means of a Wheatstone Bridge arrangement. The galvanometer was made
of a convenient sensitiveness for the purpose ; and only when a very steady tempera
ture was obtained during a set of observations was the galvanometer in a steady
enough state of approximate balance to render measurements possible. It was for the
purpose of further reducing the disturbances due to changing temperature that two
anchorring coils were used, with equal lengths of nickel wire as the cores, and with
the same magnetizing or merely heating current flowing through both double coils of
copper wire.
* Vol. xl. pp. 535545.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 3). 7
40
PROFESSOR C. G. KNOTT ON MAGNETIZATION
The nickel wires in these coils will be called L and M for ease of reference ; and the
same letters may, if occasion arise, be used each to designate the corresponding
anchorring coil as a whole.
The wires L and M, with their stout nickelbar terminals, formed two of the con
tiguous branches of the Wheatstone Bridge, the other branches A. and m being also
made of nickel wire, so as to minimise the possibility of thermoelectric effects in the
circuit. The resistances in B. A. ohms of the various branches at 13° C. were as
follows, L and M being the resistances of the parts of L and M included in the
magnetizing coils.
Wire.
L
M
X.
/*
Lo
M
Resistance .
31237
3041
32413
31556
2957
2925
The lengths and number of turns of the four magnetizing copper coils were as
follows : —
Length of copper wire
Number of turns
Lcoils
Inner. Outer.
899 cm.
172
1151 cm.
172
Mcoils
Inner. Outer.
992 cm.
171
1237 cm.
171
In the experiments to be described it was the nickel wire L which was studied ;
hence, to reduce the magnetizing current in amperes in the coil L to fields in magnetic
C.G.S. units we must multiply by 7*52.
The magnetizing current was measured on a Kelvin graded galvanometer, which
was calibrated in situ by comparison with a Kelvin ampere balance.
The two anchorring coils L and M were enclosed in a porcelain vessel, through the
side of which the various terminals were led. A quantity of asbestos wool was packed
round the coils so as to reduce as far as possible the convection currents of hot air when
the vessel was heated. The heating was effected by means of one, two, or more Bunsen
burners, according to the temperature aimed at. After several hours' heating the
temperature of both coils became fairly steady ; and when the galvanometer indicated
that a sufficiently steady temperature was reached, the necessary observations began
to be made. The temperature of the wire L at any instant was indicated with great
accuracy by the resistance of the wire, an independent experiment upon a wire cut
AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 41
from the same piece giving data by means of which the measured resistance could be
reduced to centigrade degrees. It is known # that the resistance of nickel changes
rather curiously with temperature, of which more hereafter. In the present case the
resistance temperature graph is, roughly speaking, a long sloping S form of curve, and
can be represented for interpolation purposes very approximately by three straight
lines. Thus, to reduce the measured resistance r of the wire (3 ohms at 15°) to
temperatures the formulse are
t = 567r  1633 from 15° to 200°
t = 339/'  227 ,. 200° „ 350°
t = 958r  6925 „ 350° „ 400°
2. The Method of Experiment. — The resistance changes due to magnetization
were measured by deflections on a delicate galvanometer after the bridge was approxi
mately balanced, a steady current being supplied from a secondary cell in the battery
branch. The galvanometer was calibrated in the following simple way. For any
particular condition the current i through the galvanometer is given by the formula
~Di = (LfjL  MA)e ..... (1)
where e is the electromotive force in the circuit, and D the wellknown expression
involving the resistances of the six conductors making up the Wheatstone Bridge. Its
value is
D = BG(L + M + A + fi) + B(L + A)(M + fi) + G.(L + M )(A + /*) + LM(A + /jl) + A/*(L + M) ,
where B is the resistance in the battery branch and G that in the galvanometer branch.
In the present experiments B was 6*2 ohms and G was 1 73 ohms.
The calibration of the galvanometer in each experiment was effected by making a
known small change in A ; and the magnetic effect on resistance gave a slight change in
L. We must therefore consider how the quantity D changes with slight changes of
A and L. Differentiation of D gives
d ® = BG + B(M + /x) + (G + /*)(L + M) + LM
^ = BG + B(M + /x) + (G + M)(A + /x) + V
The particular values of these quantities will vary from experiment to experiment
according to the temperature of the branches L and M. For temperatures higher than
the temperature of the air, L was always at a higher temperature than M, since it lay
lower down in the porcelain vessel. To obtain the approximate balance on the Wheat
stone Bridge m was appropriately varied so that the quantity (L/x  MA) rarely differed
more than l/400th from the value of Lm or of MA. When the difference was l/200th the
spot of light was driven off the galvanometer scale. In the following table the values
* See my paper on the Electrical Resistance of Nickel at High Temperatures, Trans. R.S.E., 1886 ; also a paper
by W. Kohlrausch, Wiedemann's Annalen, vol. xxxiii., 1888.
42
PROFESSOR C. G. KNOTT ON MAGNETIZATION
of the various resistances and of the quantities referred to above are given when the
temperatures of L are about 15°, 180°, 300°.
Resistances
■when the Temperatures
Branches.
of L are
15°
180°
300°
1)
6 2
62
62
G
173
173
173
L
312
6
10
M
304
5
8
A
324
324
324
H
3 16
27
26
D
5712
10099
20586
dD/dk
888
1264
2344
dD/dL
899
964
1417
Iii the calibration experiments the slight change in the resistance of X was always
effected by putting in a resistance of 30 ohms in multiple arc with a small part of X
whose resistance was  5125. That is to say, the change d\ was a decrease of 0'008603
ohms, and d\/\= —0002655.
Returning now to equation (l), namely,
D* = (VMA)e ..... (1)
we get, in consequence of the small change d\, the equation
Ddi + idB =  UedX ..... (2)
Similarly, when by application of the magnetizing field L is changed to L + £L, we find
mi + i8D = + fieSL ..... (3)
Substituting for i its value as given in (1) we get for (2) and (3)
/ La  MX dD\ 1
m.«bl(m + V dx)\
ns . sm / L/aMA d,D\
y
(4)
If, at the beginning of the experiments corresponding to equations (2) and (3), the
bridge were accurately balanced, the current i would vanish and we should have
or
di
87
_ M d\ _
L dX
A 8L
ZL
Si d\
L
di X
(5)
This was the case in many instances, and in many others the value of i was so
small as to make (Lm  MX) less than the thousandth part of L/m or MX. Under these
AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 43
conditions, taking the largest value of the ratios dD/Dd\ and dD/Ddh, namely,
89/572, we find
D* =  ,/a(m + MX 5T _)
D8i= +^^5too),
showing at once that the second terms in the brackets are negligible, and that equation
(5) still holds.
Finally, consider the most extreme case of all, in which the applied field was so great
as to produce a change dh, which made the spot of light travel from end to end of the
scale. It was necessary in this case to disturb the balance on the bridge, so that the
initial current i produced a deflection 3 '5 times that due to the imposed change d\.
That is to say, since the change d\ meant an alteration of 0*002655 in the value of L/«
or MX in the experiment corresponding to equation (2), the initial value of i in this
extreme case of the experiment corresponding to (3) was such as to make
L/xMX = 35 x 002655 L/x = 0093 L M
and
MX = 9907 L^ .
Thus equations (4) become
Bdi =  eMdk I
T>U = + e/*8l/l  0093 ^ ^) 
This extreme case occurred only at the ordinary temperature of 15° C. Putting in the
corresponding values of L, D, and dJ)/dJj, and taking the ratio, we find
di __M.dk 1
Ji ~ J SL 1  0093 x 312 x 0158
= _ ^ ** 1 4 x 9907 =  h '/ T A x 995 .
X SL 1  00458 k SL
Hence equation (5) is in error by 0*5 per cent. ; and it will be noticed that this is due
mainly to the factor by which we pass from the ratio M/V to L/X. The effect of the
second terms in the brackets of equations (4) is in all cases negligible. Hence in every
case we may write
di _ _ M dk
Si /j, SL
and in the great majority of cases use the more convenient form (5).
•
3. Eeduction of the Observations. — The results embodied in Table A were
plotted on a large scale, the change of resistance dh being plotted in terms of the field.
In any one series of experiments the temperature varied a little throughout ; but it was
easy to apply slight corrections by graphical interpolation so as to obtain a series of
isothermal curves. From these curves the values of cfiL were read off for the fields
2, 4, 6, etc., up to 34, and were then divided by the appropriate value L of the re
sistance of the nickel wire included in the coil. This was assumed to have the same
44
PROFESSOR C. G. KNOTT ON MAONKTIZATTON
Tablk A.
Containing the reduced results immediately deducible from the individual observa
tions of the experiments described above, and arranged in order of date, H
being the magnetic force in the heart of the anchorring coil, t the temperature
of the nickel coil at the instant, as determined by the measured resistance of the
tvire L, and dL being the change of resistance due to the magnetization.
H
t
dL
H
t
dL
H
t
dL
(Feb. 13
)
(Feb. 16
>
(Feb. 18)
336
14
00298
322
1791
00262
326
1275
00291
262
153
222
258
1831
207
258
130
236
218
163
176
212
185
183
212
1281
198
178
175
142
125
1808
110
161
124
152
135
187
96
89
179
79
102
1215
88
94
192
48
68
1769
51
78
117
54
71
189
25
5
1763
25
51
1132
20
52
169
11
32
176
6
32
1111
5
35
163
24
(Feb. 19)
(Feb. 21)
(Feb. 23)
308
2415
00173
304
317
00044
34
167
00296
248
2427
150
244
322
20
266
197
238
207
2421
133
202
322
25
218
204
194
153
241
107
149
3197
21
166
187
126
123
2405
88
119
322
19
130
181
96
101
239 5
76
10
323
14
108
17
68
7'1
2377
54
74
3235
11
8
163
31
49
2361
27
48
3235
5
58
153
14
36
2348
16
42
146
6
(Mar. 12;
(Apr. 1)
(May 6)
318
279
00117
318
328
00043
318
2993
00097
214
2783
96
216
328
28
316
3034
71
129
2763
69
106
3264
6
157
3027
49
89
2746
51
(May 10)
103
301
38
59
2727
34
304
3423
46
71
300
3
36
271
23
220
3443
15
45
299
2
23
2691
6
48
3422
7
27
299
08
(May 14)
•
(May 15)
(May 18)
32
179
00331
333
677
00301
341
163
00280
178
166
130
262
686
241
268
187
225
141
154
109
215
688
195
22
18
186
13
153
93
165
672
152
163
147
127
128
163
91
13
66
111
128
154
90
106
158
67
107
649
77
108
15
66
10
155
54
84
635
53
81
147
33
8
IB']
34
61
61
26
47
144
7
65
14 1
19
41
623
11
44
138
5
34
107
216
603
60
635
6
81
206
AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES.
45
ratio to the whole resistance L at all temperatures. It is possible, however, that with
the strongest currents, whose heating effect is quite apparent in the slight rise and then
fall of temperature during one set 01 readings as shown in Table A, the part of the
nickel wire included within the magnetizing coil might be slightly higher in tempera
ture than the short parts outside the coil which joined the nickel wire to the stout
nickel terminals. This would make the ratio L /L a little greater in the highest fields,
so that the quantity c?L/L would be a little smaller. It is clear, however, that any
small error due to this cause will not materially affect the broad conclusions to be
drawn from the experiments.
In Table B are given the final reductions, each column corresponding to a particular
temperature, and each horizontal row to a particular field. The numbers entered are
the increments of resistance per 1 00,000 ohms.
Table B.
Shoiving resistance changes per 100,000 ohms of nickel tvire in
various longitudinal fields and at various temperatures.
Magnetic
Resistance Changes at Temperatures
Field.
15°
65°
125°
180°
240°
280°
300°
328°
342°
34
1040
SI 6
621
475
253
141
86
45
5
32
970
770
594
450
244
138
83
44
30
901
729
565
126
235
133
81
43
28
825
679
529
401
226
128
79
41
4
26
749
629
498
375
216
123
76
38
24
682
579
463
350
205
119
73
35
22
613
529
425
324
19',
113
69
32
2
20
543
479
392
298
179
100
66
29
18
474
429
356
270
166
100
61
24
16
411
379
3 1 7
242
151
93
57
18
16
14
341
318
275
215
135
84
52
12
12
268
255
231
186
112
75
48
6
10
192
187
181
158
103
64
42
4
1
8
119
124
123
121
85
52
37
3
6
53
63
65
70
56
38
30
2
4
17
23
21
23
27
27
19
1
2
3
5
5
5
55
5
4
...
The numbers in the last column under temperature 342° C. were just measurable ;
anything under 3 is, in fact, barely outside the errors of observation.
4. Discussion of the results. — These numbers give two sets of graphs — namely,
the isothermals showing the relation between magnetizing force and resistance change
at the various temperatures, and the isodynamics showing the relation between the
46
PROFESSOR C. G. KNOTT ON MAGNETIZATION
resistance change and the temperatures in the various fields. These sets of curves,
marked b, a, are given in the accompanying Plate.
The first obvious result is the diminution of the resistance change in the higher
fields as the temperature rises. Thus the effect in various fields at temperature 15° is
from 200 to 300 times the effect at temperature 342°. So rapid is the final drop above
300° C. that we may safely regard the effect as practically nonexistent at temperature
350° C. It is just at this temperature that nickel loses its strong magnetic properties,
RESISTANCE CHANCES ACCOMPANYING MAGNETIZATION OF NICKEL
AT HIGH TEMPERATURES
.005
Temperature
200°
the permeability being practically unity. Thus we learn that the change of resistance
of nickel wire due to the application of a longitudinal magnetic field is mainly a
function of the magnetization or induction in the material, and not of the magnetizing
force.
In fields below 5, there is first increase of the resistance change as the temperature
rises. In fact, all the isothermals from 65° to 300° begin above the isothermal of 15°,
and then cross it as the field increases. This is particularly well marked in the case of
the isothermals 65°, 125°, and 180°. This phenomenon may be connected with the
fact that, up to a certain limit, the induction curve for nickel rises more abruptly
and reaches its ' wendepunkt ' in lower fields the higher the temperature. In other
AND RESISTANCE OF NICKEL WIRE AT HIOH TEMPERATURES.
47
words, the first effect of rise of temperature is to increase the permeability in lower
fields, probably because of the greater ease with which the molecular groupings
assume new configurations. But anything which tends to increase permeability must
tend to increase the effect on resistance. As the magnetization approaches its satura
tion value, rise of temperature diminishes the permeability, and rapidly so as the critical
temperature of 350° is approached ; and very similar is the effect of rise of temperature
on the change of resistance due to a given field.
The isodynamic curves indicate the existence of a further peculiarity which declares
itself at or near the temperature of 180° by a kind of cusplike peak in the graphs of
the higher fields. This peculiarity is also well brought out by calculating the differences
between the resistance changes corresponding to the successive temperatures in the pre
ceding table, and dividing these by the change of temperature. These average differ
ences per degree will correspond to the mean of the extreme temperatures ; and their
values for five of the fields are given in the following subsidiary Table C.
Table C.
Showing differences per degree calculated: from Table B.
Magnetic
Differences per Degree at Temperatures
Field.
40°
95°
152°5
210°
260°
290°
314°
335°
34
45
33
27
37
28
27
15
29
28
29
25
23
29
25
25
14
26
20
13
15
17
20
18
20
13
14
10
ooi
ooi
04
09
10
IT
1:4
02
6
 002
 0003
09
07
045
04
TO
From these few examples we see that there is at or near the temperature 200° a
peculiarity which shows itself by an increase in the rate per unit rise of temperature at
which the resistance change due to a given field is diminishing.
5. Comparison with results formerly obtained. — When we compare the results
here given with those obtained in the earlier experiments a considerable discrepancy
declares itself. From the results given in the final table in the earlier paper (Trans.
R.S.E., vol. xl. p. 543), we readily find by interpolation the resistance changes at the
three temperatures 127, 57"5, and 93'5, corresponding to the fields 30, 22, and 14.
Then, from the table given above (p. 45) we can interpolate values corresponding to
the same fields and temperatures. These are compared in the following short table (D),
the earlier and later results being distinguished by the Roman numerals I. and II.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 3).
48
PROFESSOR C. G. KNOTT ON MAGNETIZATION
Table D.
Field.
30
12° to 15°
I. II.
57
I.
5 "5
II.
93
I.
'•5
II.
349
901
376
754
405
647
22
215
613
233
543
260
477
14
109
341
121
321
140
297
When this discrepancy was noticed, the first idea was that some fundamental mis
take had been made in reducing one of the series of experiments. The mistake might
have been made either in calculating the field in C.G.S. units or in calculating the
changes of resistance.* It is important to make sure that no such mistake has been
made. I shall therefore reproduce exactly the observed numbers in two distinct
experiments, one made on 19th June 1902 with the first apparatus, the other made on
13th February 1903 with the second apparatus. Anchorring coil I. had 846 turns,
with a mean radius of 3 "6 cm. ; and anchorring II. had 344 turns, with a mean radius
of 9*1 cm. Hence the reducing factors which enable us to calculate field from current
are in the ratio of about 6*2 to 1. Now, in the two experiments mentioned above,
the measured deflections of the currents on the Kelvin graded galvanometer were,
respectively, 9*9 with the magnet at mark 44'35, and 22 with the magnet at mark 16 ;
that is, the currents were in the ratio of 44*35 x 22 to 9'9 x 16 ; that is, about 6'3 to 1.
Hence in these two experiments the fields were nearly the same, somewhere in the
neighbourhood of 26. The following are the readings as jotted down in the experi
mental notebook, with the deflections and differences immediately deduced from them.
Experiment I. — Field = 26*5.
( 1 ) Calibration of galvanometer.
Shunt in \.
inf.
20
inf.
20
inf.
Reading on Scale when Current is
Direct. Reversed. Direct.
134
166
1333
165
1327
173
1412
174
142
1745
134
1657
1331
165
1326
Deflection.
+ 39
247
+ 408
23
+ 418
Difference of
Deflections.
648
643
6455
* The constancy of the resistances of the various coils in use and the steadiness of the results obtained in all cases
quite preclude the possibility of any error in the estimation of the constants arising from faulty insulation.
AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES.
(2) Measurement of resistance change due to applied field.
49
Direction
of Field.
Deflection when Field is
Off. On. Off.
Difference.
+
113
1698
1144
561

117
173
1193
554
+
121
176
1209
55
122 1779
1225
555
mean
= 55*5
Experiment II. — Field = 26*2.
(l) Calibration of galvanometer.
Shunt in A.
Reading 01
Direct.
1 Scale when Current is
Reversed. Direct.
Deflection.
Difference of
Deflections.
inf.
134
1627
134
287
30
169
1278
1685
+ 4095
6996
inf.
1335
163
1337
294
30
169
1277
169
+ 413
7053
inf.
1339
1629 1338
 2905
mean
= 7025
Magnetic
Field.
off
on
off
on +
off
(2) Measurement of resistance change.
Reading on Scale when Current is
Direct. Reversed. Direct.
Deflection.
1505
2398
1415
2315
1368
1465
575
1559
645
160
149
2347
140
229
136
+ 3
+ 179
 15
+ 165
 23
•75
•75
•25
•75
•6
Difference of
Deflections.
1855
1862
mean = 185 '9
In the calibration experiments conducted as described on p. 41, the column
headed ' shunt ' indicates that in the second and fourth lines the resistance X was
slightly altered in value by joining in multiple arc a fairly large resistance and a small
part of A. In Experiment I., 20 ohms resistance was joined up with a resistance of
0"3085 ; in Experiment II., resistance 30 was joined up with resistance  5125. The
50
PROFESSOR C. G. KNOTT ON MAGNETIZATION
corresponding changes in A* were 00909/203015 and 02627/305125. Thus we
find by equation (4) : —
in L
in II.
rlL/L =
0909
555 x 2
203015x3089 ' 64"45
2627 1859
= 0025 ;
305125 x 32413 70"25
= 00703
Notice that in I. (1), II. (l) and (2), the deflections are really double the true values,
whereas in I. (2) the deflection is given at once. For in the last case the approximate
balance is altered by the magnetizing force being put on. In the other cases the
deflections are due to the reversal of the current supplied by the single cell in the
battery branch of the Wheatstone Bridge.
The two cases here given in detail prove that there can be no doubt as to a
difference of effect under apparently similar magnetic conditions. The nickel wires
used in the experiments were cut originally from the same piece of wire. The only
difference between the two forms of apparatus lay in the manner of winding. In the
first small anchorring the nickel core was a small compact closelywound coil of twenty
windings of silkcovered wire ; in the second large anchorring the nickel core was a
looselywound coil of some 10 or 11 turns, with asbestos paper interwoven. It is possible
that in the compactlywound coil the inner turns were screened from the full magnetic
action of the applied field by the outer windings. This view receives some corrobora
tion from the manner in which the discrepancy established by the figures given above
diminishes as the temperature rises. Taking the ratios of the corresponding changes
in II. and 1. we get the following results : —
Ratio of Resistance Changes (II : I) at
Field.
12°15°
57°5
93°5
30
258
201
160
22
285
233
183
14
313
265
212
Thus the measured effect in the earlier experiment deviates more from the same
effect in the later experiment the lower the field and the lower the temperature. But
this is j ust what would be expected if the discrepancy were due to magnetic screening,
which is well known to become less evident in higher fields. There are no experiments,
so far as 1 am aware, as to the effect of temperature on the screening effect ; but
we have every reason to expect that it will diminish as the temperature rises. The
In the earlier paper X was called n ; its value was less than the value of A. in the present paper.
AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 51
question here raised would probably repay further experimental investigation ; and it
obviously suggests a new method for studying magnetic screening.
6. Comparison with other peculiarities of nickel at 200° C— That some
kind of peculiarity should occur at about this temperature was not unexpected.
It was indeed with the expectation of getting some such effect that these experiments
were originally planned more than a dozen years. It was my good fortune as
an undergraduate to assist the late Professor Tait in the thermoelectric investiga
tions which occupied his attention during the years 18724.* Probably the
most remarkable results established by these investigations were those in connection
with iron and nickel. The thermoelectric lines for all metals save iron and nickel are
approximately straight through great ranges of temperature. Their inclinations in the
properly constructed thermoelectric diagram give the Thomson Effects in the corre
sponding metals. In every case of a pure metal except those mentioned, the Thomson
Effect retains the same sign throughout. In the case of iron and nickel, however, it
changes sign — at a dull red heat in the case of iron, and at about 180°200° in the
case of nickel. But the nature of the phenomenon is the same in both. The Thomson
Effect, which is negative at ordinary temperatures, becomes positive at higher
temperatures ; and finally, when the temperature is raised still higher, negative again.
The second change of sign occurs in each case at the temperature at which the metal
ceases to be strongly paramagnetic. In the case of iron, another phenomenon is known
to occur at the temperature of dull red, namely, the sudden expansion during cooling
discovered by Gore, and the accompanying reglow discovered by Barrett. No
similar effect has been observed in the case of nickel, possibly not because it does not
exist, but because the temperature is too low to admit of a visible ' reglow.' In any
case these phenomena point to a curious molecular change occurring both in iron and
in nickel at a temperature well below that at which the magnetic permeability becomes
unity.
Mention has already been made as to the rather curious manner in which the
resistance of nickel changes with temperature. In my paper on the electric resistance
of nickel at high temperatures, referred to above, this peculiarity is established, and
the difficulty in working at high enough temperatures prevented me establishing the
existence of the same peculiarity in iron, although there was indication of its existence.
This, however, was done shortly afterwards by W. Kohlrausch. The peculiarity in
the case of nickel is shown by the interpolation formulae given above, p. 41. The
rate of increase of resistance with temperature undergoes a sudden increase at a
temperature of about 180°200° C, and then diminishes as abruptly again at about
400° C. Once again, then, we have another set of phenomena indicating a peculiar
molecular change in nickel at 200° as well as at 400°.
In the present investigation the relation that is being studied involves the measure
* See Tait, Trans. Roy. Soc. Edin., vol. xxvii. pp. 125140; also Scientific Papers, vol. i. pp. 218233.
TRANS. ROY. SOC. EDIN., VOL. XLL PART I. (NO. 3). 9
52 PROF. KNOTT ON MAGNETIZATION AND RESISTANCE OF NICKEL WIRE.
merit of small changes, which are indeed changes of the second order, namely, the
change per unit rise of temperature of the change due to a given applied field. It
would be utterly impossible, in the present state of knowledge regarding molecular
groups, to make any prediction as to how the molecular change indicated by the
thermoelectric and resistance peculiarities should show itself in the present case. A
glance, however, at the isodynamics, with their cusplike jpoint in the higher fields,
and a tendency to a maximum in the lower fields, seems to indicate some peculiarity
at this critical temperature of 200°. What seems to be indicated is, that about this
temperature the change of resistance with magnetization begins to fall off more
quickly as the temperature is raised.
( 53 )
IV. — The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D. (Edin.),
M.A. (Camb.), F.G.S., Lecturer in Geology at the University of St Andrews.
Communicated by Professor James Geikie, LL.D., F.P.S. (With a Plate.)
(Read February 15, 1904. Issued separately April 9, 190:1.)
CONTENTS.
PAGE
I. Introduction 53
II. Previous Literature 54
III. Physical Features and Geology of the
District 56
IV. Description of the Deposits
1. Tlie Lower Boulder Clay
2. The Sands and Gravels
3. The Upper BoulderClay and Rubbly
Drift ... . .
V. The Boulders and Erratics
VI. General Conclusions
page
63
63
68
74
77
82.
I. Introduction.
The area embraced in this paper consists of that part of Pembrokeshire which
lies to the north and northeast of St Bride's Bay. Bounded on the west by St
George's Channel and on the north by Cardigan Bay, it extends to the northeast as
far as the mouth of the river Teifi, near Cardigan.
That part of the country which lies in the immediate neighbourhood of St David's
has, through the laborious researches of the late Dr Hicks and others, become well
known to geologists, and may now be regarded as classic ground. The solid geology
of this promontory has given rise to much discussion, and has, perhaps, attracted more
attention than that of any other part of the Principality. The reason for this great
interest is to be sought in the facts that the rocks of this area are of a very great
antiquity, and that the sedimentary series contain the remains of some of the earliest
organic forms yet found in the earth's crust, whilst the igneous rocks are also displayed
in great abundance and variety, and present us, in the words of Sir Archibald
Geikie, with "the oldest wellpreserved record of volcanic action in Britain."
The geology of the district lying immediately to the northeast of the St David's
promontory has not been the subject of so much attention, but the investigations
carried on by De la Beche and the other officers of the Geological Survey before the
middle of last century have recently to some extent been revised by Mr Cowper Eeed,
and his results are published in a paper entitled "The Geology of the Country around
Fishguard," which appeared in the Quart. Journ. Geol. Soc. (vol. li., 1895, p. 149).
But while so much has been written concerning the ancient rocks of this country,
very little attention has been paid to the more recent geological deposits. Owing to
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 10
54 DR T. J. JEHU ON
the facts that geographically the region lies further south, and that it does not present
such a great elevation of the land above sealevel, it is not to be expected that
Pembrokeshire will show such marked traces of the former presence of glacial conditions
as are to be met with in North Wales. Nevertheless, it has been known for a long
time that this region is to a large extent covered by more or less loose and uncon
solidated material, which is usually spoken of as Drift. And references are found
scattered in the geological literature of the district relating to travelled boulders and
other possible glacial phenomena there seen. But hitherto no attempt has been made
to give a connected description of the glacial deposits with a view to the unfolding
of the sequence of events which occurred during and after the Glacial epoch, and of
correlating the results obtained by an examination of this area with those derived from
a study of glacial deposits in North Wales and other regions. The need for further
investigation will be evident to anyone who compares the map (plate i.) in Professor
James Geikie's work on The Great Ice Age, illustrating the British Isles during the
Epoch of Maximum Glaciation, with the late Mr Carvill Lewis' " Sketch Map of
England and Wales showing the Edge of Land Ice," which is reproduced in Professor
Bonney's Ice Work. In the former the southern boundary of the great icesheet
is made to pass beyond Wales and run along the Bristol Channel ; and the northern
ice which overwhelmed Anglesea is marked as crossing the western end of the Lleyn
promontory of Carnarvonshire, and, joining the Irish Sea, it fills up St George's
Channel and crosses the extreme tip of Pembrokeshire at St David's Head. In the
latter the landice is shown as not extending over the whole of South Wales to the
Bristol Channel, but with its southern edge extending no further south than is indicated
by a line drawn eastwards from the St David's promontory, and the glaciation of
Northern Pembrokeshire is attributed solely to local ice — the northern ice apparently
extending no further south off the Welsh coast than the Lleyn promontory.
The results obtained during the investigations carried on by the present writer will
at any rate serve to settle the dispute with regard to the southward extension of the
Northern or Irish Sea Glacier.
II. Previous Literature.
References to the surface deposits and surface features of Pembrokeshire are meagre
and scanty in the extreme.
Sir R. Murchison, in The Silurian System (p. 520), makes the following remarks :
' The detritus which appears on the surface of most parts of Pembrokeshire is of a
simple character and, as in other parts of South Wales, is of local origin. It consists
of fragments of greenstone, porphyry, carboniferous grits, etc., all of which can be
traced to the various mountains forming the crest of the country. In some parts this
detritus is exceedingly coarse. ... In other tracts, as north of Haverfordwest, we meet
with finely comminuted gravel ; but this is rare."
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 55
Symonds, in his book entitled Records of the Rocks (p. 53), refers to the fact
that the country near St David's must in former years have been largely covered by
boulders. These have now to a large extent been cleared away from the surface and
used for building rough walls which serve the purposes of fences and hedges. He adds
that in his opinion " these boulders are all local, and have travelled over a slope of
ice and snow which once reached from the Trap Hills of Precelly down to the sea."
And in another place (p. 181) he speaks of the Precelly hills in North Pembrokeshire
as being " studded with icecarried boulders, which were used for cromlechs and tumuli
by a prehistoric race of men."
Messrs Howard and Small, in their " Geological Notes on Skomer Island," which
appeared in the Trans. Cardiff Naturalists' Soc. (vol. xxviii., 1896), assert that distinct
evidences of the action of ice are seen on the mainland opposite Skomer. And on
Skomer Island itself blocks are found which have travelled from the St David's district
and some possibly from North Wales. Flints were also seen scattered about.
Professor M'Kenny Hughes, in his paper " On the Drifts of the Vale of Clwyd "
(Quart. Journ. Geol. Soc, vol. xliii., 1887), remarks that "the lowlying plateau at St
David's is covered by a gravel containing flints." But he found no traces of shells there.
In a paper by Professor Bo^ney " On the Socalled Diorite of Little Knott
(Cumberland), with further Remarks on the Occurrence of Picrite in Wales" (Quart.
Journ. Geol. Soc, vol. xli., 1885), some observations are found communicated by Dr
Hicks relating to the glaciation of the St David's region. A boulder of picrite was
found on the promontory just to the east of Porthlisky, " resting immediately on
Dimetian rock, surrounded by an uncultivated area overgrown by gorse and heather."
The striae along the coast are said to run usually from northwest to southeast. He
adds : " But it is clear that very many of the boulders scattered over it must have come
from the high land in the northeast of Pembrokeshire, the Precelly range. There is
ample evidence of local till, and in places (at considerable elevations) of marine sand
with transported boulders, fragments of flint being common among them." Dr Hicks
was of opinion that " this points to the derivation of some of the materials, including
possibly certain boulders, from a northwest source."
The most important communication which has appeared on this subject is a very
short report read by Dr Hicks at the Cardiff meeting of the British Association in 1891,
" On the Evidences of Glacial Action in Pembrokeshire, and the Direction of IceFlow."
This report is reproduced in the Geol. Mag. for that year. He there refers to the
presence of icescratched rocks and of northern erratics in the district. The direction
of the glacial striae and the probable presence of erratics from North Wales and from
Ireland " would tend to the conclusion that glaciers from these areas coalesced in
St George's Channel, and that the ice which overspread Pembrokeshire was derived
from both these sources, as well, probably, as from a flow extending down the Channel
from more northern areas." By far the majority of the boulders are said to be of local
origin, but he notes a large boulder of granite and another of picrite found on Porth
56 DR T. J. JEHU ON
lisky farm. " The picrite boulder has been shown by Professor Bonney to resemble
masses of that rock exposed in Carnarvonshire and Anglesea, and the granite boulder,
which before it was broken must have been over 7 feet in length and 3 to 4 feet in
thickness, is identical with a porphyritic granite exposed in Anglesea, but not found
anywhere in Pembrokeshire." He found clear evidences showing that St Bride's Bay
was overspread by a great thickness of drift from the hills immediately to the north.
" The intervening preglacial valleys were also filled by this drift, and the plains and
rising grounds up to heights of between 300 and 400 feet still retain evidences of its
former presence, and many perched blocks." Chalk flints were found at heights of
over 300 feet, and have probably come from Ireland. He refers also to the crushing
and bending of the strata at places, and to some wellmarked examples of " crag and
tail," but he does not locate these phenomena.
The late Professor Prestwich, in his paper on " The Raised Beaches and ' Head '
or RubbleDrift of the South of England" (Quart. Journ. Geol. Soc, vol. xlviii., 1892),
refers to the possible occurrence of this rubbledrift on the coast of Pembrokeshire. He
thought that he had detected traces of a raised beach and ' head ' near Porth Clais,
and again at Whitesand Bay.
Professor Bonney, in his Ice Work (p. 161), states that " In Pembrokeshire and the
adjoining districts erratics are often abundant, as may be seen near St David's. At
present no systematic attempt has been made to trace them up to their sources, but
they have probably come from the higher ground inland, that is to say, roughly, from
the northeast." And again (p. 165) he refers to the possibility that the northern ice
travelled down the bed of the Irish Sea, and perhaps ultimately overflowed St David's
Head.
Dr Wright, in his book on Man and the Glacial Period, remarks that "At St
David's peninsula, Pembrokeshire, striae occur coming in from the northwest, and,
taken with the discovery of boulders of northern rocks, may point to a southward
extension of a great glacier produced by confluent sheets that choked the Irish Sea "
(p. 143).
Mr Cowper Reed, in the paper already referred to, mentions the fact that "drift
or boulderclay causes a difficulty in tracing the boundaries or determining the
characters of the underlying beds " in the Fishguard district.
III. Physical Features and Geology of the District.
The county of Pembrokeshire lies in the extreme southwest corner of the
Principality, and that part of it which is under consideration in this paper extends
further to the westward than any other part of England and Wales, with the exception
of the extremity of Cornwall. The promontory of St David's is washed on three sides
by the sea which has eaten into the land so as to give rise to a variety of recesses and bays.
It is the presence of hard igneous rocks that has enabled it to resist the ceaseless action
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 57
of the waves, which, owing to the direction of the prevalent winds, often beat upon the
coast with great fury. The softer Lower Palaeozoic slates and Carboniferous shales to
the south have succumbed to the encroaching sea, and there given rise to the broad and
wide bay of St Bride's. The islands lying to the west consist partly of hard igneous
rocks, and no doubt were once joined to the mainland. The coast scenery is magnificent,
and throughout this region the rocky cliffs rise steeply out of the sea, and sandy beaches
are only found here and there, such as at Whitesand Bay, Abereiddy Bay, Abermawr,
Goodwick Bay, and Newport Bay. The cliffs in places are nearly perpendicular, and
everywhere exhibit excellent sections of the rocks. The outline of the coast is very
jagged, reefs and stacks of rock sticking out here and there, whilst, on the other hand,
the sea has penetrated in so as to form caves and coves, and small narrow rialike
channels, such as that seen at Porth Clais at the mouth of the Alan river, and that at
the mouth of the Solva river. A siltedup estuary occurs at Abermawr to the west
of Strumble Head, and a larger one on the east side at Goodwick Bay ; they now form
swampy ground. The only other estuaries of importance lie on the northeast side of
Strumble Head, at the mouths of the rivers Gwaen, Nevern, and Teifi. That of Gwaen
is also somewhat Walike in character.
On the north coast Strumble Head is a prominent feature and stands out boldly
to sea, and a little further to the northeast a small but wellmarked headland occurs at
Dinas.
The oneinch Geological Survey maps of this part of Wales were prepared before the
end of the first half of last century, and no revision has yet been made. Since that
time a more complete knowledge of the fossil contents of the sedimentary series has
been obtained, and improved methods for the study of igneous rocks, especially with
regard to their microscopic structure, have been introduced. The need for a fresh
survey is generally recognised, but much new light has been thrown on the geology of
St David's promontory during the last thirty years through the researches carried on
by Dr Hicks, Sir A. Geikie, and others. And Mr Cowper Reed has examined and
described within recent years the geology of the area around Fishguard. But the
region is a very complicated one, and much of the geology still remains obscure. The
rocks of this part of Pembrokeshire are almost entirely of Lower Palaeozoic age, and a
remarkable variety of both the igneous and sedimentary kinds is there displayed. In
the St David's region a very full development of Cambrian rocks is exhibited, and these
are underlain by a series of volcanic rocks — both series often showing signs of meta
morphic changes. The base of the Cambrian was taken by Dr Hicks to be marked by
a conglomerate in which are enclosed pebbles of the underlying rocks. The volcanic
tuffs and breccias which underlie the conglomerate were taken to be preCambrian.
Underneath these again comes a granitoid mass, which he regarded as still older.
Later, the district was visited by Sir A. Geikie, who, after an examination of the ground,
arrived at the conclusion that the granite is an intrusive mass, and that there is no
break between the Lower Cambrian rocks and the volcanic series underlying them.
58 DR T. J. JEHU ON
The granite covers an area lying immediately to the south of St David's, and there
is another wedgeshaped mass a little to the southwest, reaching the coast on the
eastern side of Porthlisky. The St David's mass graduates into a spherulitic quartz
porphyry and felsite at its northern end. The granite is surrounded on all sides but the
south by rocks of the volcanic series which are marked as Andesites on the indexmap.
These form a ridge running E.N.E. and W.S.W., stretching from Llanhowell, past the
city of St David's, to reach the coast at the southern end of Ramsay Sound. Two
detached masses are marked further east, about Llanreithan.
The volcanic group consists largely of bedded tuffs ; but lavas also occur, and give
rise to prominent crags to the west of St David's.
Dykes and sheets of diabase traverse the other formations.
These igneous rocks are flanked on the west and southwest by the Cambrian
conglomerate, and this is followed by green, purple, and gray flaggy sandstones, with
intercalated red shales. Towards the base, fragments of Olenellus were found by Dr
Hicks. In a southeast direction these are followed by the gray and black flagstones
and shales of the Menevian series, and these again by gray and bluish flagstones and
slates of the Lingula Flag series. Still further eastwards, a small tract of Tremadoc
beds is found near Tremainbir.
Beds of the Menevian and Lingula Flag series also occur at the south of White
sand Bay. The Lingula flags run as a continuous band from the bay inland as far as
Cruglas, and on the north a band of Tremadoc flagstones and earthy slates runs parallel
to a point south of Abereiddy Bay.
The north coast from St David's Head to Abereiddy is made up of slates and
shales and flagstones of Arenig age. Masses of gabbro occur at St David's Head, and
a little east of this, diabase masses, giving rise to rugged eminences, are seen.
Northeast of Abereiddy the Llandeilo flags succeed the Arenig series, and consist
of black slates and flags, sometimes calcareous, and some felspathic tuffs.
Numerous bands of " felspathic trap " are seen to occur in the tract bordering
Abereiddy Bay.
Eastwards from the St David's promontory, right into mid Wales, the sedimentary
rocks are marked in one colour on the Survey maps, and are referred to as " Lower
Silurian (including Upper Silurian not yet separated)." They consist of shales, slates,
and gritty sandstones, with some flagstones. Mr Cowper Reed found that beds of
Llandeilo and Bala age form Strumble Head. Immediately around Fishguard we
meet with Llandeilo and Arenig beds, and further east with Upper Llandeilo and Bala
beds again. Dinas Island is composed of sandstones, slates, and conglomerates of
Llandovery age.
The country around Fishguard is rich in igneous rocks. Felsites, tuffs, and
agglomerates contemporaneous with the Llandeilo and Bala beds occur, and intruded
into these are sills and masses of " greenstone." The latter include basalts, dolerites,
diabases, and gabbros.
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 59
Contemporaneous volcanic rocks and intrusive sheets of diabase occur in the district
"between Fishguard and Newport, and further south in the Precelly mountains strips of
felspathic rock are indicated on the Survey maps.
East of Newport the igneous rocks die out.
Blown sands are heaped up at places on the coast, and are generally full of land
shells. To the east of Whitesand Bay they rise to heights of 150 feet and are pro
longed inland for nearly a mile, giving rise to the tract known as " The Burrows " or
" Towyn." They are also seen at Abermawr, and cover an extensive area at Newport,
and again at the mouth of the Teifi, where they form dunes known as the " Towyn
Warrens."
Towards the western end the country, as seen from a height, presents the aspect of
a flattened tableland or plateau, having for the greater part of its extent no great
elevation above sealevel, but here and there having rocky knobs and masses jutting
out, especially towards the western and northwestern extremities of the promontory.
The view, as seen from one of the hills on Ramsay Island, suggested to Sir Andrew
Ramsay that in this part of Wales we have the remnant of an old plain of marine
denudation, which is continued into Cardiganshire and further inland. The rugged
masses of Cam Llidi and Pen Berry, which rise so boldly just east of St David's Head,
and have relatively the appearance of considerable mountains, only attain heights of
595 feet and 576 feet respectively. Further east the rough eminences which stand
out so prominently on Strumble Head are mostly under 600 feet in height — Garn Gelli
alone exceeding that limit and attaining 625 feet. Garn Fawr, famous for the remains
of ancient fortifications there found, is just short of 600 feet high.
Between Fishguard and Newport is a ridge of high land, reaching elevations of over
1000 feet at Mynydd Melyn, Mynydd Caregog, and Cam Ingle. To the south of
these rise the Precelly mountains, which have a somewhat smooth outline and attain
heights up to 1500 feet. From Newport to Cardigan the country is hilly, but not
mountainous — the highest point being Pen Creigiau, which is 642 feet above sealevel.
The highest ground is formed of diabase and other intrusive igneous masses ; the
volcanic rocks occupy ground above the average level, while the lowlying ground
consists of the more easily denuded slaty and shaly beds.
Passing from west to east, the land generally becomes more hilly, and the higher
grounds from Newport to Cardigan are composed of hard sandstones or arenaceous slates.
The main streams of the district occupy preglacial valleys, and have cut their way
through the drift which once filled them. The estuaries of the rivers Solva, Alan, and
the Gwaen are Walike in appearance, and it is probable that an arm of the sea once
extended for some way up the lower course of each ; for the estuaries are trenchlike,
with steep rocky walls on either side for a considerable distance inland, and it is hardly
conceivable that these have been cut out altogether by the action of the streams.
On the northern coast there are two peninsulas of a very peculiar character, for
they are separated from the mainland by trenchlike valleys, which, though now never
60 DK T. J. JEHU ON
occupied by the sea, look as if they had been so in comparatively recent times. They
are both spoken of locally as islands : one lies between Abereiddy and Porth Gain, and
is known as Barry Island, and the other is Dinas Island, west of Fishguard. The
valley between Dinas and the mainland is particularly striking, being only a few feet
above sealevel, whilst that at Barry Island is not much less noteworthy, though its
bottom attains a somewhat higher level. These peculiar valleys, together with the
indications shown of the former presence of the sea up the inlets at the mouths of many
of the streams, and the occurrence of swampy estuaries such as those seen at Abermawr
and Goodwick, seem to point to a slight rise of the coast within recent times, causing a
retreat of the sea.
But at the present day the sea seems to be gaining once more on the land. At
several places along the coast peat is seen at low tide, and in most of the bigger bays
evidences of buried forests are sometimes seen. Giraldus Cambrensis, who wrote in
the twelfth century, says with regard to St Bride's Bay at Newgall : — " When Henry II.
was in Ireland an unusually violent storm on that sandy coast blowing back the sand
discovered the appearance of the land concealed for so many ages ; stumps of trees
standing in the sea, with the marks of the hatchet as if done but the day before, a very
black earth and wood like ebony, so that it appeared not so much like a seacoast as a
grove."* And George Owen, in The Description of Pembrokeshire, written in 1603,
gives an account of a very similar occurrence. He says : " About xij or xiij yeeres
past were seene on the sandes at Newgall, by reason as it seemeth the violence of the
sea or some extreeme freshe in the winter, washed awaye the sandes (w ch dayelye is and
was overflowen with the tyde), soe lowe that there appeared in the sandes infinitte
nomber of buttes and trees in the places where they had been growinge, and nowe
euerye tyde overflowen : there appeared the verye strookes of the hatched at the fallinge
of these tymber, the sandes being washed in the winter, the buttes remained to be seene
all the sommer folio win ge, but the next yeere the same was covered againe with the
sandes : by this it appeareth that the sea in that place hath intruded upon the lande." t
Again about sixteen years ago a big storm washed away the sand and exposed roots of
great trees in Whitesand Bay. Huge logs of oak trees were carried away by the
neighbouring farmers, some of which are still stored, and were shown to the writer.
Twigs and branches of hazel were found in abundance, although no hazel grows
now near St David's. The writer is also informed that horns of deer were picked
up
Similar evidence of a buried forest has been discovered in Goodwick Bay, and all
along the coast up Cardigan Bay. All this reminds one of the old Welsh tradition
regarding a great inundation of a land called Cantref Gwaelod, situated in the region
now covered by Cardigan Bay, which is usually attributed to the fifth century. There
is also a very old tradition that St Bride's Bay was formed by an inrush of the sea.
* Rolls, ed. vi. 100. t Page 247 in the Cymmrodorion reissue.
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 61
Referring to some of the shallow valleys and peculiar bays seen in Pembrokeshire, Sir
Andrew Ramsay, in his essay " On the Denudation of South Wales and the adjacent
Counties of England," contributed to volume i. of the Memoirs of the Geological Survey t
says (p. 329) : "In numerous instances valleys opening to the sea end in bays, and it
will be found that not infrequently the headlands on either side of these bays were
composed of comparatively unyielding materials. Depress the land, and these valleys
become arms of the sea ; raise it, and the bays become a continuation of the valleys.
If Pembrokeshire were elevated but 60 or 70 feet, Milford Haven would become a
shallow valley of this nature, with occasional pools or small lakes in its hollows, through
which would wind the water now flowing into the haven." And similarly, if the county
was depressed but 60 or 70 feet, the valleys at the loAver part of their course would form
seainlets like Milford Haven, and Barry and Dinas would be separated as islands from
the mainland. It is practically certain that movements of elevation and of depression
have taken place within comparatively recent times, but it is a difficult problem to
ascertain the extent and duration of these movements.
The land of the St David's promontory was long ago described by Giraldus
Cambrensis as a " stony, barren, unimprovable territory, undecked with woods,
undivided by rivers, unadorned with meadows, exposed only to winds and storms."
Since his time the land has yielded somewhat to the continuous treatment of genera
tions of farmers, and a great part of it is now under cultivation. But much rough
uncultivated moorland still remains, which in places is overgrown with gorse, and
sometimes shows a boggy nature. Such are the commons seen round about St David's,
and also further east. Parts of them are occupied by shallow sheets of water, as at
Trefeithan common and Dowrog common. As a rule these commons are of a clayey
nature, and in places some peat formation is seen. Peat becomes more evident in the
country lying to the south of Fishguard and southwest of the Precelly hills, and it is
dug even right up on the Precelly hills themselves. The country from Newport to
Cardigan is well cultivated for most part.
Throughout the area under consideration in this paper much of the soil is of
a distinctly sandy nature. Hence the land is very dry and needs much rain. It is an
old saying in this part of Pembrokeshire, that " in summer rain every day is too much,
and every second day too little."
The greater part of the land is covered by a blanket of superficial material, which
may all be included under the name of Drift. This is somewhat variable in character,
but near the surface a sandy element seems to predominate. However, as traced
laterally, the sand often passes abruptly into clay or clayey loam, and vice versa. This
drift near its upper part is usually stuck full of boulders and rockfragments of all
kinds, and of all shapes and sizes. Good sections are seen along the coast in some of the
bays, but it is very rarely that one meets with a good exposure inland. Smoothed,
polished, and icescratched boulders can be picked out of the drift in plenty, and occur
throughout the district. It is more difficult to meet with examples of striated and
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 11
62 DR T. J. JEHU ON
smooth rocksurfaces, and this is partly owing to the fact that so much of the
country is covered by the superficial accumulations ; and the igneous rocks which
project above the surface are so weathered and worn from exposure for such a great
length of time, that any marks of glaciation which they may once have shown must
have become almost entirely obliterated.
Reference has already been made to Dr Hicks' observations on the striae seen on the
rocksurfaces at places along the coast near St David's. The best example noted by the
present writer is at AVhitesand Bay, just at the northern corner. Here the rock, as it
appears from under the cliff of drift, presents a distinctly hummocky appearance, and is
smooth, polished, and well striated — the striae having a course from northwest to south
east. It is interesting to note that out of the drift above, striated stones can be readily
picked : these are generally subangular, with blunted angles and somewhat smooth
and rounded edges. In the small valley coming down near Porth Melgan, and separating
the rocks of St David's Head from Cam Llidi, a pavement of sedimentary rocks has been
exposed by the removal of some turf, and this pavement shows distinct marks of
glaciation. Another very good example of a glaciated rocksurface is seen quite at the
other end of the district at Gwbert, on the coast to the north of Cardigan. Here,
emerging from beneath the drift again, a smooth and striated rock is seen — the direc
tion of the striae being from a little west of north to a little east of south, showing that
the ice must have come on to the land from the region of Cardigan Bay.
The igneous rocks of St David's Head and those lying further east, especially at Pen
Berry, appear to be somewhat moutonne on their northern aspect, but no unmistakable
glacial striae were seen, and this is no doubt due to the fact that the rockfaces are so
much weathered.
On the greater heights there is a general absence of perched blocks and big erratics.
It is quite possible that perched blocks may have been common in former times, but
they have in all probability been removed by man, for the region is full of traces of
defences prepared by primitive man, and these usually take the form of great collections
of boulders and stones gathered together and heaped up in the form of dykes. Splendid
examples of such ancient entrenchments are seen near St David's Head, and again on
Strumble Head. The few blocks seen on the high ridges are almost invariably of the
same nature as the underlying and surrounding rocks.
Southwest of St David's, in the Treginnis tract, some huge boulders are seen. A
big one lies on the hill above the cliffs at Penmaenmelyn, but it consists of a somewhat
coarse andesitic rock, which is found in situ at no great distance away. The boulders of
granite and of picrite found near Porthlisky farm by Dr Hicks have been already
mentioned. The whole country was once strewn with boulders ; and although many
still remain scattered over the land, most have been cleared away and used for building
dykes, etc. An examination of the stones in the dykes shows that they are almost
entirely of rocks found in the locality, as might be expected. Boulders of the St
David's Head gabbro are found carried in a southeastern direction, and are plentiful in
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 63
the neighbourhood of Caerfai Bay and Caer Bwdy Bay. In the country between St
David's and Strumble Head boulders are also common. Blocks of "greenstone" are
found around Mathry : the rock is a diabase, probably occurring at no great distance away.
On Strumble Head the boulders are very plentiful, and the writer found a boulder of
picrite not far from the extreme head, near the coastguard station, and another further
south, near TreSeissyllt, together with a remarkably fresh olivinegabbro, of a kind
which is not found in the district. These will be referred to more fully in another
section of the paper. East and south of Fishguard, boulders of a dark "greenstone"
and of a volcanic rock which weathers white are abundant. They are found on the hill
slopes, on roadsides, and in the fields, but many have been cleared away as before.
The Precelly hills are free of boulders as compared with the hills lying further north.
It may be mentioned that vestiges of antiquity abound in this part of Pembroke
shire. Hut circles, ancient entrenchments, cromlechs, British camps, etc. occur at
places all the way from St David's Head to Cardigan, being particularly evident at St
David's Head and Strumble Head. Some of the ancient British towns or settlements,
traditions of which are to be found in the old Welsh romances, called The Mabinoonon,
are situated within this area, and one has been identified as occurring on the Garn
Fawr, Strumble Head.
IV. Description of the Deposits.
The glacial deposits of the district vary a great deal as traced laterally from
place to place. Owing to the want of good exposures inland, it is generally impossible
to mark out the limits of the different kinds of superficial detritus. One has to
depend for most part on a study of the sandpits and clay or marlpits which are
dug in places all over the district. But it is not often that these go down to any great
depth ; and when occasionally a deep pit has been dug out for the purpose of obtaining
clay or sand for the land or for building, it is invariably filled up again with surface
rubbish, so as not to be a danger to animals. Where a pit appeared to be of peculiar
interest, the writer employed a man for digging, and in this way obtained some
valuable sections. At the eastern limit near Cardigan there are brickworks, and here
it is that the best sections are to be seen. The writer hopes in the near future to con
tinue his investigations eastwards in the neighbourhood of Cardigan. The deposits,
which occur in the district are the following : —
3. Upper BoulderClay and RubblyDrift.
2. Sands and Gravels.
1. Lower BoulderClay.
1. The Lower Boulder Clay. — This is a typical boulderclay which is met with
in patches throughout the district, but is best and most fully developed towards the
east. It has received no attention within recent times, but a very quaint and, on the
whole, a very accurate description of it is found in the works of a writer who lived in
<U DR T. J. JEHU ON
the time of Shakespeare. In an article on Sir Roderick Murchison's Silurian System
in the Edinburgh Review, 1841 (vol. lxxiii. p. 3), it is stated that " one of the oldest
inquirers connected with the geology of this ancient region is George Owen of Henllys,
in Pembrokeshire, who has been called the patriarch of English geologists." This
worth} Welshman left behind him a manuscript work on the topography of his native
country — a book of great value and interest. It was published in the Cambrian
Register, 1793, and has recently been reproduced, under the editorship of Mr Henry
Owen, in the Cymmrodorion Record Series. The book has been already referred to, and
is entitled The Description of Pembrokeshire.
His observations on the boulderclay are so good that they are well worth quoting.
Writing of " the naturall helpe and amendementes the soil it selfe yealdeth, for
betteringe and mendinge the lande," he refers to what he calls " Claye Marie." " This
kind of Marie is digged out of the Earthe, where it is found in great quantitie, and
thought to be in rounde great heapes and lompes of Erthe as bigg as round hills, and is
of nature fatt, toughe, and Clamye. . . . The opinion of the Countrie people where this
Marie is founde is that it is the fattness of the Earthe gathered at Noes flood, when the
Erthe was Covered withe the said flood a whole yeare, and the surginge and tossinge of
the said flood, the fattness of the Earth being clamye and slymie of nature did gather
together, and by rowlinge vpon the Earthe became round in forme, and when the flood
departed from the face of the earthe, the same was left drie in sondrie partes, which is
nowe this Marie that is found, and how the Common people Cam to this opinion I
knowe not, but it is verye like to be true, for wheresoever the same is founde, it is
loppie (loose) and covered with sande, gravell, and round peblestones, such as you shall
flncle at the sea side verie plaine, appearing that the stones hath ben worne by the sea
or some swift river."
" Also in the harte of the Marie is founde diverse sortes of shells, of fishe, as Cogle
shells, Muskell shells, and such like, some altogether rotten & some yet unrotted, as
also you shall therein finde peaces of tymber that ben hewen with edge tools & fire
brandes, the one ende burned and diverse other thinges which hath ben before tyme
vsed, & this XX tie foote and more deepe in the Earth in places that never haue been
digged before, and over the which great oakes are now growinge ; and this seaven or
eight myles from the sea, so that it is verie probable that the same came into these
places at the tyme of the great and generall flood "
" This marie is of couler with vs most commonlie blwe and in some place redd."
" It is verie hard to digg by reason of the toughness, much like to waxe : and the
pickax or mattock beinge stroken into it, is hardlie drawne out againe, so fast is it
holden : it is alsoe verie heavie as ledd." " This Marie is founde in Kernes and both
Emlyns from Dynas vpp to Penboyr in Carmerthen sheere, beinge about twentie myles
in lengthe and about fowre myles in bredeth in most places to the sea syde, and out of
this compasse I cannot heare that the same ys founde : I thinke more for want of
Industrie than otherwise " (pp. 71, 73). He ends up his remarks on the Clay Marie
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 65
thus : — "And, who so list to learne more of this Marie : let him pervse a pamphlett
which I have written thereof, wherein I have declared the nature of the marie, how to
know yt and finde yt, and the order at Lardge of digginge and layeinge yt on the
lande ; of the severall sortes thereof for what yt is good, and for what yll. And so for
brevyties sake I Cesse to writte any More thereof." It is a great pity that this
pamphlet has never been published, for it would be of great interest to geologists, as
perhaps the earliest attempt to give a full description of the boulderclay. It seems
that the treatise was finished in 1577, and consists of twelve chapters. A footnote to
the preface of Mr Henry Owen's edition of his Pembrokeshire (p. xxiv) states that a
copy of the pamphlet lies in the Vairdre Book at Bronwydd, " written out of a copy
in his own hand, by me, John Owen of Berllan, 1684."
The Kernes mentioned in the above extract is that part of Pembrokeshire lying
between Dinas and Cardigan.
The present writer can bear testimony to the general accuracy of the description
given by George Owen. It is at the brickworks, Cardigan, that the best exposure
is seen. There in the pit a vertical section of this boulderclay 20 feet deep is seen.
It is darkbluish in colour, but after drying becomes more of a light bluishgray. It
is a remarkably tough and tenacious clay, and can only be dug with great difficulty, for
no crevices or fissures are seen and no trace of bedding. The whole mass is strikingly
homogeneous and uniform in character, and has evidently been subjected to great
pressure. The bottom is not reached in the section, and so the depth attained by it
at this place is not known. For most part it is very free from stones, but a little
further east in the same pit these are rather more commonly met with. Some beautifully
glaciated subangular and blunted boulders were seen, with the striae running princi
pally in the direction of their longer axes. Many of these boulders are of Carboniferous
Limestone, and these interfere very much with the manufacture of the bricks, and are,
as far as possible, picked out by the workmen. Boulders of conglomerate, grit, shaly
and slaty rocks were also noted, and many of igneous rocks, which are foreign to the
district. These will be dealt with again below, in another section.
One of the most characteristic features of this Lower BoulderClay is the presence
of marine shells scattered irregularly through it. They seem to occur chiefly in its
upper part, and are invariably much broken and worn, and therefore very difficult to
identify. The fragments are also extremely friable. Occasionally small waterworn
pebbles of quartz, etc. are seen in the clay ; but most of the stones included are ice
worn rather than waterworn in the Cardigan pit. Another striking feature is the
presence of fragments of woody matter in the clay, sometimes at a depth of 15 to 18
feet.
Above the boulderclay in the brickyard occurs 2 or 3 feet of sand and gravel and
a yellowish stony clay, and towards its north end this stony clay increases in thickness
to at least 7 feet, passing in places into yellow sand.
66 DR T. J. JEHU ON
At Cardigan the height of the boulderclay above sealevel is under 50 feet.
Between Cardigan and Dinas this blue clay is seen in patches underlying small tracts of
moorland, and it attains a height of nearly 600 feet a little south of PenCreigau, where,
at a short distance below the road to Cardigan, it may be seen, though the exposures
are very poor.
Just southwest of Dinas, near the roadside, claypits occur on RhosIsaf showing
a depth of 6 feet. The same stiff, compact, bluish boulderclay is here seen, full of com
minuted shellfragments. Boulders are fairly common, mostly iceworn and scratched,
but some water worn. One example of a slaty rock showed not only fine striae but
a wide groove smoothed out by ice action. The clay gets darker as traced downwards,
but the bottom is not seen. Workmen stated that it reaches a depth of at least 1 5
feet, and occasionally a thin seam or stratum — no more than half an inch in diameter —
of fine gravel is said to occur. But no trace of bedding occurs in the clay. It is
capped for 2 feet by a yellowish clay with boulders. The pits are 240 feet above
sealevel. Small exposures are seen in some of the fields on Dyffryn farm, about a
mile south of Goodwick. Owing to drying and weathering, it is of a light bluishgray
colour, and is here full of fragments of the local lavas and tuffs. Most of this farm
is underlain by this clay. A little further south, in a field belonging to Drim farm, is a
small pit of a similar character. No shell fragments were to be seen in these exposures.
Similar tough bluish boulderclay is seen in claypits on Tregroes moor, at a height
of over 250 feet. In fact, in all the moors lying to the south of Fishguard, with the
exception of those found on mountainslopes, this boulderclay can be found, but it
is unfortunate that there are no deep pits or good sections to be seen. But the
engineers of the Great Western Railway are making borings in this neighbourhood for
a tunnel, and they very kindly supplied the writer with all the information in their
possession which might be of interest. A boring has been made at Trebrython farm,
to the southwest of Tregroes moor, and about 150 yards from the railway. In the
boring the following succession was obtained : —
5 feet of earthy clay.
5 feet of yellowish clay, with rock fragments.
10 feet of a somewhat tough greyishblue clay.
Slate rock.
The greyishblue clay is the Lower BoulderClay, and it is seen that it only attains
a thickness of 1 feet here. The yellowish clay may be partly the bluish clay weathered,
but most probably consists for most part of the equivalent of the Upper BoulderClay.
The boring is made at a height of nearly 300 feet above sealevel. About three
quarters of a mile down the railway towards Goodwick is a railway cutting passing
through part of the superficial deposits, and a boring has been made here also. A full
section of this cutting and boring is given on another page, but it may be mentioned
here that typical tough blue boulderclay is there shown which attains a depth of
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 67
18 feet, and is followed above by Sand and Upper BoulderClay. The Lower Boulder
Clay is partly exposed in the cutting, and is full of fragments of marine shells.
There is also a small exposure in the moor just south of Letterston.
That the boulder clay occurs far inland is proved by an examination of a pit at
Llyn, near Llangolman, not far from Maenclochog. When seen the pit was only 5
feet deep, but it is opened up by the farmers periodically to a depth of from 15 to
20 feet, and ladders have to be used to get in and out. The clay is bluish in colour
and very tough, and is usually spoken of by the natives as " indiarubber clay."
It occurs in boggy land at the bottom of a small valley, at a level of over 400 feet.
On both sides the land rises to a higher level, and is covered by a mantle of sands and
gravels.
West of Goodwick the blue clay comes to the surface just below St Nicholas Church,
at Clyn Bach moor. The depth was not ascertained. Fragments of broken shells
occur here, though somewhat rarely. Some boulders of igneous rocks, foreign to the
district, were found, and among them an unmistakable boulder of the Ailsa Craig
riebeckite microgranite.
The bluish clay is seen at places as far as the western end of the promontory — such
as in the moors around Trefain, and in the shallow pits at Henllys, below Llanrian,
where very fine examples of chalk flints were seen included. In the pits of Trefeithan
moor, west of St David's, this boulderclay is exposed to a depth of 6 feet and is of the
usual character. No shells were seen, but some vegetable matter occurs in the clay.
The boulders were few, and those noted were all of local rocks. Somewhat similar
but shallower pits may be seen on Dowrog moor, Tretio moor, and Caerfarchell
moor.
It is thus evident that this Lower BoulderClay occurs throughout the district, but
it appears to thicken as we pass from west to east, and to be best developed to the
east of Strumble Head. Fragments of marine shells were found in the clay at Cardigan,
at Dinas, at the railway cutting near Tregroes, and at St Nicholas, and for the deter
mination of these and other shells mentioned in this paper the writer is indebted to
Mr Henry Woods, M.A., St John's College, Cambridge. Owing to their fragmentary
condition it has been difficult to identify with certainty the shells found in the clay.
But the following species appear to be represented : —
Cardigan Claypit —
Pectunculus glycimeris, L.
Astarte sulcata, Da Costa.
Mytilus, sp.
Dinas Claypit —
Astarte (Nicania) compressa, Mont.
(?) Cyprina islandica, L.
68 DR T. J. JEHU ON
Railway Cutting (between Tregroes moor and Manorowen) —
Pectunculus glycimeris, L.
Cardium islandicum 1 Chem.
Vulsella modiola 1 L.
Astarte sulcata, Da Costa.
Astarte (Nicania) compressa, Mont.
Venus (Ventricola) carina, L.
Cyprina islandicu, L.
Some of the boulderclay from the boring near Tregroes was washed and examined
under the microscope. A good deal of very fine sandy material — mostly quartz — was
observed, and a few foraminifera could be seen.
2. The Sands and Gravels. — Above the Lower BoulderClay comes a series of aqueous
deposits, consisting of sands and gravels, which are sometimes stratified and sometimes
show hardly any traces of stratification. These deposits vary very much in thickness,
and are apt to die out suddenly when traced laterally. They usually occupy a higher
level than that attained by the Lower BoulderClay, and are often seen banked on the
lower slopes of the hills. In places where sections are seen passing through the
different deposits, no gradual passage can be traced from the lower stiff blue clay to
the sands and gravels above — and this suggests that the sands and gravels lie upon an
eroded surface of the clay. The sands are as a rule yellowish and yellowishbrown in
colour, and have all the appearance of being marine : they are very variable in texture,
and show all gradations from very fine sand to coarse gritty sand and gravel. And
the gravels are often coarse and pebbly, resembling the shingle collected on beaches.
At some places the sands are charged with worn and broken fragments of marine
shells : these usually occur more abundantly in the fine gravel or coarse gritty sand
than in the fine sand, although minute flakes can often be detected in the latter. They
seem to be collected together in the stratified beds at certain spots, and to be absent
in somewhat similar beds exposed only a short distance away. For instance, shells are
plentiful in the Manorowen sandpit, whilst not a trace of shells can be seen in the
Cnuc sandy pit, which lies only about a quarter of a mile further south : and 50 yards
or so beyond the Cnuc sandy pit shellfragments are again found in sand exposed in
the railway cutting. In the pits where shellfragments are found it may often be
noticed that small pieces are cemented to the surface of a rounded stone. This is
doubtless due to a deposit of carbonate of lime derived from the decay of some of the
shells.
Like the Lower BoulderClay, the deposits of sand and gravel become better
developed when traced from west to east, and the best sections and pits can be seen in
that part of the district lying to the east of Strumble Head. Though the St David's
promontory is largely covered by a loamy sand, no sections were seen showing deposits
of the wellmarked marinelike sands and gravels found further east. At Tyllwyd
there are small pits reaching a depth of 5 to 6 feet, where the loamy sand is well
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 69
shown. The bottom of the deposit is not seen, but as traced downwards the material
becomes very sandy. It is ferruginous, and has a reddish colour.
The sands and gravels become more evident in the neighbourhood of Mathry. A
little to the east lies a rugged mass of igneous rock known as Y Graig, and on the
south side of this there is a newly dug pit, exposing 8 feet of pebbly gravel and
gritty sand, with streaks of fine yellow sand. The deposit gets more sandy as traced
downwards, but the bottom is not seen. A tendency to a rough bedding is shown in
the section. At Pont Duan, north of the roadside, a very similar gravelpit is seen.
Pen Cnuc, at Castle Morris, marks the site of a mound of fine yellow sand, most of
which has now been carried away.
Gravel and sand is exposed, but only to a depth of 4 feet, in a field 200 yards north
of Bridgeend, and again at Heathfield, south of the house. Further north, Tregwynt
lies on sand and gravel, and much sand is seen between Tregwynt and Trellys.
East of St Nicholas Church the sands and gravels cover most of the land, as may be
seen in pits in many of the fields. The writer employed a man to dig here in order to
ascertain if possible the depth attained by the sand and gravel. At 8 feet the bottom
was not reached, but the gravel became wet, and it is probable that the blue clay lies a
few feet lower down, for it crops out in the moor below St Nicholas Church. The
pebbles in the gravelpit were all well rounded, and chalkflints were seen. Gravels
and sands seem to cover much of the ground on the Strumble Head promontory, but
at places a yellowish earthy clay replaces them at the surface, though they may here
occur with the clay. In a boring made for a well at Llandruidion farm sand was
brought to the surface, in which comminuted shells were seen. The boring reached a
depth of over 20 feet, and rock was not reached.
In the farmyard at Trehowell, near the northern extremity of Strumble Head, in
sinking for a well, no sand was passed through — all was earthy clay ; but in a field 250
yards further north a sandpit occurs, where 2 feet of fine yellowish sand are seen,
covered by 2 or 3 feet of a loamy and somewhat stony clay. Mounds of gravel and
sand occur on Caergowil, on the heights above Goodwick, and sections 7 feet deep are
exposed. They are of the usual character.
South and east of Goodwick and Fishguard deposits of sand and of gravel are
frequently met with. They do not occur in the form of kames or eskers, but are found
lying in the slopes of the minor hills, and sometimes spreading to the top. It would
be almost impossible to map them, as their occurrence is so irregular and patchy ; they
are apt to die out laterally in a sudden way, passing into clay or rubblydrift. At
many places they are overlain by an Upper Boulder  Clay. Perhaps the most
interesting of all the sections is one seen in the Manorowen sand  pit, which lies
in a small wood on the roadside, immediately south of the farm buildings, at a height
of nearly 200 feet above sealevel, and two miles distant from Goodwick Bay. When
visited a section of only 5 feet was seen, but means were taken to deepen it down
to 12 feet. A somewhat diagrammatic view of this section is shown in fig. 1.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 12
70
DR T. J. JEHU ON
The materials were well stratified, and occur in the following order from above
downwards : —
A. Soil and rubblydrift 13 feet.
B. Very fine lightbrown sand, with the beds somewhat contorted, passing down
into thin beds of darker sand, followed by coarse sand with some pebbles.
Near the bottom is a thin band of sandy clay, 3 inches . . . . 4£ feet.
C. Pebbly sand, variable in thickness . . . . . . . 1 to 1J feet.
D. Coarse grey sand or line gravel, showing bedding. Some layers are more
distinctly pebbly, and here and there fine sand occurs. The fine gravel is
full of fragments of marine shells ........ 6 feet.
Bottom not reached.
1llt , u w j*ittobd**~ Jil '**«*ta aiJ _
Soil and rubbly
drift.
Fine sand, show
ing some fold,
iug.
Sand with peb
bles.
~  — «. — » Pebbly sand.
Coarse grey sand
and fine gravel
with shell frag
ments.
Greatest length 1 2 feet.
Depth about 12 feet.
Fig. 1. — Diagrammatic Section of the Manorowen Sandpit.
Many of the shells have been identified, and are discussed below. Chalkflints are
common. On the opposite side of the road, below the churchyard, and at a lower
level, there is a small exposure — 3 to 4 feet deep — which consists entirely of a coarse
gravel ; but above, towards Manorowen Hill, the gravels and sand are replaced at the
surface by clay.
About a quarter of a mile further south, sand is seen again at Cnuc Sandy. There
is a big pit just in front of the cottage, 8 feet deep. What is seen here is for most part
very fine yellow sand. Gritty and gravelly streaks and layers occur here and there,
dying out as traced horizontally. No traces of shells were found here. The pit was at
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 71
one time 15 feet deep, and the bottom of the sand was not then reached. 1 to 3
feet of stonydrift cap the sandy beds at the surface.
At a distance of 50 yards further south the railway cutting has passed through
1 feet of similar very loose sand, in which fragments of marine shells occur plentifully.
The town of Fishguard is, in part at any rate, built on sandy deposits, and a good
exposure is seen in a quarry on the roadside going down Fishguard Hill to
Goodwick Bay. It consists of yellowish sand and fine gritty gravel of the usual kind,
which near the top becomes more of a loamy, stony drift. North of the town gravel
pits are common in the fields, and at Pwll Landdu on the coast, east of Castle Point,
the cliff is largely made up of gravel and a ferruginous sand, capped by a yellowish
boulderclay full of stones.
A little north of the valley of the Gwaen, at Trellan, near Llanllawer, fine yellow
sand occurs on the lower slope of the hill of Ceunant.
It would be useless to mention every spot where the sands and gravels are to be
seen. They occur in patches all the way to Cardigan, being especially well seen
in LlwynyGwaer Park.
The highest level attained in this part of Pembrokeshire by the sands and gravels
is at Pen Creigiau Cemmaes, just off the road leading from Nevern to Cardigan, and
four miles distant from the latter place. Sand occurs at the top of the hill, at an
elevation of 640 feet. Most of the hilltop is evidently of sandy material, and in a pit
a section 8 feet deep is seen, showing very fine yellowish sand passing downwards into
darker and more gritty material. There is only a faint trace of bedding. The bottom
is not seen. This spot is nearly three miles distant from the coast. Chalkflints and
wellrounded pebbles of quartz are found. A few yards down the hill on the northern
side are other small exposures, about 4 feet deep, showing more pebbly sand with
rounded boulders ; and on the southern side, immediately below the main road to
Cardigan, is a gravelpit, in which are seen rounded and subangular stones, some a foot
in length. Chalkflints and pebbles of white quartz were common, and a boulder of
Millstone Grit and of a reddish granitoid rock foreign to the district were picked up.
Also two pebbles of a muscovite granite. These will be referred to again below. A
rough kind of stratification could be seen — layers of small gravelly pebbles separating
beds of coarse shingle. The pit is 8 to 10 feet deep.
At Pantgwyn, half a mile north of Pen Creigiau, sand is seen in a pit, and it is
darker and more gritty than that on the hilltop.
Deposits of material resembling marine sands are met with even north of Cardigan,
as at Bancywarren, but this is outside the area embraced in this paper.
Similar deposits are found far inland, even south of the Precelly hills. A few
yards south of Eosebush there is a sandpit on the western side of the railway. A
diagrammatic section of it is shown in fig. 2. The lower part is hidden by a talus
slope. Above this comes 4 to 5 feet of fine yellow sand, very ferruginous in places.
The sand becomes a little clayey or loamy in the eastern half of the section, and is
34 feet  
46 feet • — rHS
45 feet
72 DR T. J. JEHU ON
traversed by patches and imperfect layers of a blackish hard panlike material, which
is probably organic in nature. Above this comes 3 to 4 feet of rubbly material, full of
fragments of slaty and other rocks of local origin.
Fine yellow sand is seen also near Llangolman, a little east of Maenclochog, on the
slopes of the ground rising from the moor, and at Charing Cross there is a pit showing
roughlybedded gravel and sand to a depth of 8 feet. A smaller pit of a similar nature
lies at Cefn Ithyn, just north of Maenclochog.
Further south, in the neighbourhood of Trefgarn, sand of quite a different character
occurs. There is an exposure of 12 to 15 feet in a big gravelpit on the roadside
opposite the Chapelofease, near Nantycoe mill. The fine gravel and sand here is
dark grey in colour, and consists largely of minute flattened flakes. The sand is not
^ ^ ,» J » J ..» r .^^»<r. ^3'g*r^33g5 S3aB».^^ Soil.
Iff 1 !*. ^^^5?^&^^^~
Fig. 2. — Diagrammatic Section of the Rosebush Sandpit.
yellow nor brown like that found further north, which the natives speak of as
" Demerarasugar " sand. In Trefgarn Hall park there is a pit where the material
seen is somewhat similar, but much coarser. The stones are rounded and subangular.
The deposits here do not remind one so much of the sand and gravel found on sea
shores.
List of Mollusca found in the Manorowen Sand and Gravel Pit*
Lamellibranchia.
Nuculana pemula, Mull. ......... rare.
Pectunculus glycimeris, L. ........ . very common.
Barbatia lactea, L. ......... . very rare.
Mytilus edulis, L. .......... rare.
Vulsella modiola 1 L. . . . . . . . . . very rare.
Astarte sulcata, Da Costa ......... very common.
Astarte (Nicania) compressa, Mont. ....... moderately rare.
Astarte (Tridonta) arctica, Gray ( = boreal 'is) ...... moderately rare.
* The specimens which I collected from this place, together with some obtained subsequently by Mr V. M.
Turnbull, are now in the Sedgwick Museum, Cambridge.
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE.
73
Cyprina islandica, L. .
Tellina (Macoma) balthica, L.
Maetra (Spisula) solida, L. .
Venus ( Ventricola) easina, L.
Tapes {Amygdala) decussatus 1 L.
Cardium islandicuml Chem.
Mya truncata, L. .
common,
moderately rare,
rare.
moderately rare,
very rare,
very rare,
rare.
ScAPHOPODA.
Dentalium entalis 1 L. .
Gasteropoda.
Puncturella noachina, L.
very rare.
Natica clausa 1 Brod. and Sow.
.
rare.
Turritella communis, Lam.
common.
Buccinum undatum, L.
common.
Tritonofusus gracilis, Da Costa
very rare.
Ocinebra erinacea, L
very rare.
Trophon (Boreotrophon) clathratus,
L. .
common.
„ „ scalar if or mis, Gould
moderately rare
Nassa (Hima) incrassata, Strom. .
very rare.
Bela turricula, Mont.
rare.
Bela rufa, Mont, (elongate form)
rare.
Fragments of other shells were also found, but they were too broken for identifica
tion. Some of the shell fragments found were very thick, especially pieces of Cyprina
islandica. Many are rolled, and the majority very broken. Entire single valves of
Astarte compressa were found ; and of the univalve shells, Ocinebra erinacea and
Trophon clathratus occurred in nearly perfect condition.
The fauna appears to contain a mixture of species belonging to different climates :
Astarte borealis, Trophon clathratus, and Trophon scalariformis are Arctic and
Scandinavian species, not now found living in British seas.
Astarte compressa, Cyprina islandica, Buccinum undatum, and Puncturella
noachina belong to a northern type of British species which inhabit Arctic and
Scandinavian seas in common with our own.
The shells are in very much the same condition as those which have been obtained
at Moel Tryfan and at Gloppa, and most of the forms found at Manorowen occur in
the other two places also. But Nuculana, which occurs rarely at Manorowen, is
common at both of the other places.
Pectunculus glycimeris is abundant at Manorowen, but very rare at Moel Tryfan
and Gloppa. Venus easina, though frequent at Manorowen, is also rare at the
other places.
Samples of the sand from several places were examined microscopically, and they
all showed a very close resemblance to marine sands. Most of the grains were of
74 DR T. J. JEHU ON
quartz, and the smaller ones were angular, while the bigger ones tended to be more
rounded.
3. TJ'p'per BoulderClay and RubblyDrift. — The sands and gravels are in many
places covered by a yellowishbrown boulderclay, quite different in character from the
bluish boulderclay which underlies them. This Upper Boulder Clay is sometimes
fairly tough, and is generally much more stony than Lower Boulder Clay. It varies
very much in thickness and character. Inland it often only occurs as a thin covering
a few feet deep, but on the coast, where the best exposures are seen, much greater
depths are attained. Sometimes it is a tumultuous unstratifled till, with boulders of
all shapes and sizes scattered pellmell throughout the matrix. At other places — and
it may be at no great distance away — it has more of the character of a rubbledrift,
and as seen in section, has the appearance of an agglomeration of coarse and more or
less angular debris, showing a rude kind of bedding. It is evident that most of it
consists of material which has been rearranged to some extent, and afterwards modified
by subaerial agencies. It is impossible to separate the more typical unstratified
boulderclay from the rough semi stratified clayey and sandy rubbledrift. The
included boulders are derived in the main from the rocks of the district, but
many fartravelled stones are also found, and these will be discussed in the next
section. Icescratched stones are fairly common. These are usually subangular,
with blunted angles and rounded edges. Bounded waterworn stones are also common,
especially in the resorted rubblydrift. No traces of marine shells are seen in the Upper
BoulderClay and RubblyDrift. On the coast it is found capping the rocky cliffs at
places, and in the bays fine sections, sometimes over 20 feet deep, are exposed. The
foreshore is often covered with big boulders derived from the neighbouring cliffs. As
the sea is now gradually gaining on the land, the cliffs of drift on the coast are being
continually undermined, and the included stones and boulders are washed out and
strewn over the shores. The beaches on the bays are rich in boulders and stones of
rocks foreign to the district. These have undoubtedly been derived from the cliffs of
drift, which are constantly undergoing a process of degradation owing to the action of
the waves and of subaerial agencies.
By far the best exposures of these upper deposits are shown on the coastline of
the St David's promontory. Figs. 4 and 5 (Plate) represent sections seen in Whitesand
Bay, and give a very good general idea of the appearance of these Upper Drift deposits.
Fig. 4 shows a section of the cliff near the north end of the bay. Here the cliff is
about 20 feet high and consists entirely of drift. At this spot the drift is a
typical till or boulderclay, showing no bedding, but full of stones and boulders, big and
small, which are scattered confusedly through it. Most of the boulders are angular and
subangular, and some are well glaciated. A few rounded pebbles and stones also occur.
The boulders are mostly of local rocks, though some erratics are seen. Loose sandy
soil occurs at the top. At the base the rock does not appear, but the shore is covered
with stones derived from the cliff. A few yards further north slaty rock is seen
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 75
emerging from underneath the boulderclay, which becomes thinner in this direction.
At places the rock shows a hummocky surface, marked with glacial strise, which run
from northwest to southeast. At the base of the small promontory called Trwyn
Hwrddyn, on the north side, a rubble of very coarse fragments lies between the solid
rock and the drift. Here the drift shows a rough sort of stratification, and has
much sandy and pebbly material intermingled with boulderclay.
Fig. 5 shows a section in Whitesand Bay seen further south. This is also about
20 feet in height, but here it consists more of a rubbly drift. The matrix is earthy and
sandy, and is chokefull of small flakes of slaty and other rocks, which have a rude
kind of arrangement, especially towards the lower part. Boulders of larger size occur
here and there, and consist for most part of grit and conglomerate and slaty rocks,
with some blocks of the local gabbros and diabases. Pebbles of white and yellowish
white quartz are common. The top is covered by loose yellow sand, probably wind
blown, and just underneath this are traces of a pebbly bed. Though rock does not
appear at the bottom just at this spot, it crops out on both sides a short distance away.
Drift of a similar kind is seen in sections, and capping the cliffs at other places on
the western coast of the promontory.
Boulderclay is seen at Porthlisky stuck full of stones, many of which are smoothed,
polished, and striated ; and a boulder of the St David's Head gabbro, measuring roughly
3x2x2 feet, lies on the beach below. A stony boulderclay or a more rubbly drift
is seen at various places on the south coast, the best exposures being above Caerfai
Bay and at Caerbwdi Bay. At Caerbwdi the cliff is over 20 feet in height, but the
base is hidden by talus : the matrix is here rather sandy, and streaks and pockets of
rather fine sand are seen here and there. The included stones are often pebbly, but
some are subangular and icemarked. They are made up almost entirely of rocks
found in the neighbourhood. Near the top flaky fragments are very plentiful, and
these are derived from the local purple flagstones and slates. A big boulder of the
coarse gabbro from St David's Head lies at the base of the cliff.
No good section is seen at PorthyBhaw, but the drift caps the hills and cliffs
to the southeast.
Very similar sections are seen on the north coast at Abereiddy and above Traeth
Llyfn. At the latter place a rubble of big boulders is seen ; most of these are of local
igneous rock, very ironstained and decomposed. Towards the top the section is freer
of big boulders, and is full of little flakes of sedimentary and cleaved rocks.
One of the finest sections on the coast is seen at Abermawr, west of Strumble Head.
At the northern end the rock is seen capped by 10 to 15 feet of stonydrift. As traced
southwards the drift thickens to about 40 feet, then tails off rapidly. Where thickest
the lower part shows some tendency to a rough kind of bedding, and is full of small
flakes and little stones, more or less pebbly. This passes above into a rubbly clay, full
of boulders of all sizes, most of which are angular and subangular, and derived from
rocks of the locality. The cliffs on Strumble Head in many places are covered by a
76 DR T. J. JEHU ON
mantle of stony till, but good sections are rare. The sections of drift exposed
on the coast between Strumble Head and Cardigan are not so good as those seen west
of Strumble Head, but where seen they are of the usual character.
Inland, good sections are not often met with. Much of the country in the
neighbourhood of St David's is covered by drift, which consists of a sandy matrix full
of boulders. But often the matrix is clayey, as may be seen in sections on the
roadside near Castell, southwest of St David's, and again in a pit just off the road
near Pont Clegyr, two miles east of St David's. In fact a large part of the country
southwest of Strumble Head is covered by material which has been to a large extent
rearranged, and which cannot be defined accurately either as boulderclay or as sand
and gravel, though the tendency is for the sand and gravel to become more marked at
a depth of a few feet from the surface. Much rubblydrift, becoming more sandy
as traced downwards, is spread out on Strumble Head, especially on the moorland above
Goodwick. The sands and gravels occurring south and east of Fishguard Bay are, as
already mentioned, usually overlain by a few feet of rubblydrift or more typical stony
boulderclay. In the railway cutting between Tregroes moor and Manorowen 7 feet
of stiff yellowishbrown boulderclay is seen covering the shelly sand. This clay is
spread out over much of the high land skirting the railway here on the west side.
A little further south the sands and gravel die out, and the Upper BoulderClay seems
to lie directly on the Lower BoulderClay, and this occurs possibly in the boring at
Trebython already referred to, where yellowish clay is succeeded by tough bluish clay.
It is very rarely that one has an opportunity of finding all the deposits succeeding
one another in the same section, and of ascertaining the depth of each. But the engineer
of the Great Western Railway at Goodwick supplied the author with particulars of the
boring made in the railway cutting between Tregroes Moor and Manorowen, just about
Cnuc Sandy. A complete section of the railway cutting, together with the results
obtained by boring, are given on the next page. From above downwards, the deposits
passed through were —
(4) Stiff yellowishbrown clay with fragments of slaterock .... 7 feet
(3) Fine yellow sand with shellfragments . . . . . . 10 „
(2) Stiff darkblue boulderclay with shellfragments . . . . 18 „
(1) Gravel 5 „
Rock.
This section proves very clearly the presence of an Upper and a Lower BoulderClay.
These are separated here, as in many other places, by a deposit of sand. Most of the
Lower Boulder Clay is below the surface of the railway cutting. And a glance at the
section shows that before the cutting was made, the surface of the ground was covered
to a depth of 7 feet by the Upper BoulderClay. So it is quite possible that the Lower
BoulderClay spreads over a much wider area than is evident at the surface, and that
much of it is hidden by more superficial deposits. It is interesting to note that
underneath the Lower BoulderClay there lies 5 feet of gravel. On comparison this was
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE.
77
found to be very similar to underlying grit rock when this is broken up. So it
may represent material ground out of the solid rock by the movement of landice.
Further inland a good example of the Upper Drift is seen in a cutting on the roadside
between Puncheston village and the railway station. It is a stony and rather rubbly clay.
Close at hand, on Puncheston common, the Lower BoulderClay crops to the surface.
Further east, up to Cardigan, the Upper BoulderClay and Rubbly Drift are seen
in many places overlying sands and gravels or lying immediately on the blue clay.
The line of division between this Upper BoulderClay and the sands and gravels
is not so marked as that between the sands and gravels and the Lower BoulderClay.
Stole AOff to \iq .
Fig. 3. — Section at Boring No. 1 on Goodwick to Letterston (existing) Railway,
between Tregroes Moor and Manorowen.
V. The Boulders and Erratics.
The transport of boulders is of great importance as indicating the general direction
of icemovement. Throughout northern Pembrokeshire boulders may be seen scattered
over the surface, and are especially common on waste or uncultivated land. Many of
those found on the slopes of the hills have no doubt rolled from the parent rock above,
but many, though not foreign to the district, have yet been carried for some distance by ice.
Where the land is cultivated an immense number of boulders must have been
removed, but even in the fields it is common to meet with big blocks of igneous rocks,
which, as a rule, have been left in the position where they were originally stranded
in order that cattle may have something to rub against, or have been left on account
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4).
13
7S DR T. J. JEHU ON
of their immense size, though these latter are becoming gradually destroyed by blasting
operations. At many places, especially on the St David's plateau, huge standing
stones, cromlechs, or other ancient remains are seen, and it is more reasonable to
believe that the immense blocks used for these purposes were found as boulders near
at hand than that they were quarried from the parent mass, which often lies at a
considerable distance away. And this is rendered the more probable as blocks of
similar size are not infrequently seen dotted over the surface. An examination of the
stone dykes will show how plentifully boulders of all sizes must at one time have been
studded over the ground, and what a variety of rocks is represented. Towards the
western end of the area the boulders are largely found to have been dispersed from
the igneous rocks on the north coast of the promontory. And everywhere the great
majority are derived from parent masses found in the district. Boulders of diabase
may often be seen resting on volcanic or sedimentary rocks, and vice versa, proving
that there has been some transport. It has already been mentioned that blocks of
the St David's Head gabbro are found lying to the southeast at Caerfai Bay, and
Caerbwdi Bay, and on the cliffs above. This implies that there was a movement of
ice from a northwesterly direction, and it agrees with the evidence shown by the
glacial strise which are seen on the coast. Erratics are met with often in the drift
and on the shores, but the number which has been noted on the surface is not great.
A further study of the stone dykes would doubtless bring more to light
1. Erratics seen on the surface of the ground. — The detection of those about to
be mentioned is mostly due to the fact that they had been broken up by the farmers
through blasting or other agency, so as to expose fresh surfaces. Hicks states (Geol.
Mag., 1891, p. 501) that he observed many northern erratics in the St David's district.
The granite boulder which he discovered on Porthlisky farm " before it was broken
must have been over 7 feet in length and 3 to 4 feet in thickness, and is identical with
a porphyritic granite exposed in Anglesea." He found another of picrite which he thus
describes : " The boulder is somewhat rounded ; its longer axis, which lies nearly south
east and northwest, measures about a yard. A transverse section is slightly triangular,
the shorter sides measuring respectively about 16 inches and 22 inches. It lies on the
promontory forming the east side of Porthlisky harbour, resting immediately on
Dimetian rock, surrounded by an uncultivated area overgrown by gorse and heather "
(Quart. Journ. Geol. Soc, vol. xli. p. 519). It was submitted to Professor Bonney for
examination, and he states that it is wonderfully like the boulders found at Peny
Carnisiog, Anglesea, which had been previously described by him (Quart. Journ. Geol.
Soc, vol. xxxvii., 1881, p. 137). In a later paper (Quart. Journ. Geol. Soc, vol. xli.,
1885, p. 518), Professor Bonney remarks that " the lithological evidence rather favours
the derivation of the Anglesea boulders from dykes in that island." A hornblende
picrite of a somewhat similar character occurs also in situ at Penarfynydd, on the
southwest coast of Carnarvonshire. So it is probable that the boulder found near St
David's has been carried by the agency of ice from Carnarvonshire or Anglesea.
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 79
The writer discovered a big boulder of picrite on Strumble Head, on a piece of
uncultivated ground a little north of Tresinwen farm, by the side of the pathway
leading to the coastguard station at the Head. The boulder had been broken, and
now lies in several pieces — the biggest piece measuring roughly 3 x 3 x 2 feet — and
in its original condition it must have been much bigger. Another boulder of picrite,
very similar in appearance and size, and also broken, was discovered on a field lying a
little to the northwest of Treseissyllt, between the farm and the coast north of Aber
bach. A microscopic section of this rock revealed the presence of the following
minerals : — brown hornblende, strongly dichroic ; augite, nearly colourless ; olivine,
not very abundant, and always very much serpentinised and of a greenish colour ;
magnetite ; a chloritic mineral, which is evidently an alteration product, and a little
plagioclase felspar. The chloritic mineral is greenish in colour and markedly dichroic.
The general appearance of the rock as seen in section was very different to that of
the St David's picrite : the augite does not show such perfect forms in the former
as in the latter, and there is rather more olivine in the St David's rock. The
Strumble Head picrite boulder is rather more like some of the Penarfynydd speci
mens as seen in section, but there is not so much olivine in the former, and the
pcecilitic structure which often characterises the latter is not seen in the former.
But it is highly probable that the Strumble Head picrites have also been borne
from Lleyn or Anglesea. The two specimens found on Strumble Head lie about
three miles apart, in a line whose direction is northnortheast to southsouthwest.
Near the Treseissyllt boulder of picrite lay a boulder of olivinegabbro, also broken
to pieces by blasting. The newlyexposed faces were remarkably striking, and the
crystals are very fresh. The rock is quite unlike the gabbros found in Pembroke
shire. A microscopic section showed beautifully fresh olivine crystals, and the rock
is undoubtedly of Tertiary age, and has probably come either from the Western Isles
of Scotland or from the northeast of Ireland.
2. Erratics in the Drift. — As might be expected, the majority of the boulders
found in the drift deposits are of local origin. They occur abundantly in the Upper
BoulderClay and RubblyDrift, and in the sands and gravels, and to a less extent
in the Lower BoulderClay. The grits, shales, and slaty rocks of Pembrokeshire are
very similar in appearance to rocks of a like nature from North Wales, and the same
is true of some of the igneous rocks, especially the diabases and some of the lavas.
It is thus quite possible that among the boulders found imbedded in the drift many
North Wales rocks may be represented, though there is no means of distinguishing
them readily from the boulders of local origin. This was suggested to the writer by
the discovery of boulders of a diabase rock in the boulderclay exposed at Cardigan.
To the naked eye this diabase seemed very like that found to the southwest in
Pembrokeshire. But no such rock is known to occur anywhere nearer Cardigan than
Newport — nine or ten miles to the southwest — and so these boulders must have come
from the north. This is made all the more probable by the discovery of boulders of
80 DR T. J. JEHU ON
what are undoubtedly northern rocks associated with these boulders of diabase at
Cardigan.
Of the erratics the most important discovery in the drift was that of a small boulder
of the Ailsa Craig riebeckite rock, or paisanite. This was found in the bluish boulder
clay, near the surface, at ClunBach moor, St Nicholas, near the southwestern end of
Strumble Head. The specimen was sliced, and as seen under the microscope it is
identical with specimens obtained from Ailsa Craig.
Boulders of hornblendeporphyrites from the southwest of Scotland occur in the
Lower BoulderClay as far east as Cardigan, and are found even oftener in the Upper
stony BoulderClay. Several varieties are seen, all of which can be matched in Wig
townshire and Kirkcudbrightshire. But the erratics which are most commonly met
with in the drift are reddish granophyres, quartzporphyries and microgranites.
Manv of these are dykerocks, and it is very difficult to trace them to their source.
Some have certainly come from Ireland, and some most probably from the southwest
and west of Scotland. Lake District rocks and North Wales rocks are not so well
represented.
In the claypits of Bhos Isaf, near Dinas, excavated in the Lower BoulderClay, a
reddish granophyre was found, which, under the microscope, resembles very much some
of the granophyres of Mull. And in the Pen Creigiau gravelpit boulders of granophyre
occur, which have come from the Carlingford district, Ireland. One was sliced, and the
microscopic characters seen were identical with that of the granophyre of Barnavaine,
Carlingford. These reddish granophyres are found in the drift throughout the area,
but seem to be rather scarcer in the extreme west. As the writer had not much
opportunity of comparing them with Irish rocks, he sent them to Prof. Watts, of
Birmingham University, who very kindly examined them. Many of the granophyres,
he thinks, can be matched in the Carlingford mountains ; others bear more resemblance
to the Tertiary granophyres of the Inner Hebrides. The reddish quartzporphyries
appeared to him to be like the varieties of the quartz porphyry of Cushendale in
Antrim ; and among the boulders of microgranite he noted two which are likely to
have come from the mass of microgranite at Cushendun in Antrim. He is of opinion
also that the Old Red conglomerate of Cushendun might, when broken up, present
examples of many of the types of boulders found in the drift of Pembrokeshire. A
pebble of muscovitegranite, probably the Foxdale granite of the Isle of Man, was
obtained from the gravelpit at Pen Creigiau, over 600 feet above sealevel. Examples
of Millstone Grit were obtained here also, and in the Cardigan claypit boulders of
Carboniferous Limestone are common, often with the fossils well preserved. On
the beach at Gwbert, not far from Cardigan, boulders and pebbles of Carboniferous
Limestone are common. These must have come from Ireland or from carboniferous
rocks which are hidden at the bottom of the Irish Sea. It is hardly likely that they
have come from the small exposures bordering the Menai Straits in North Wales.
Chalkflints occur everywhere — in both boulderclays, in the sands and gravels, and
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE.
81
on the beaches. These also must be derived from the northeast of Ireland and
from rocks hidden under the Irish Sea.
From the cliff of Upper BoulderClay at Porthlisky a boulder of olivinedolerite
was obtained, microscopic sections of which show very fresh olivine. It is certainly
a Tertiary rock, and has probably come either from the northeast of Ireland or from
the western isles of Scotland.
3. Erratics found on the shores along the coast. — These are especially abundant at
those places where cliffs of boulderclay or drift are seen. Many of them are found
lying just at the foot of the cliffs, having only recently fallen from them ; and others
which were picked up as pebbles on the beach have doubtless, for most part, been
derived from the drift also.
Ailsa Craig, Riebeckite Rock or Paisanite . . . found on AbermaAvr beach (frequent), Aberfelin
beach, PorthyRhaw beach.
(It is interesting to note that a boulder of this was also found in the Lower Boulder
Clay at St Nicholas, not far from Abermawr.)
Granites from the Dalbeattie area, several varieties . found at Pwll Gwaelod beach (frequent), Aber
bach beach (near Dinas).
Granites of Galloway type . . . ... . ,, Whitesand Bay, Aberbach (near Dinas),
Pwll Gwaelod, Gwbert (near Cardi
gan).
,, Abermawr.
A fine specimen of a MicahornblendeGranite, identical
with that of Auchencairn, Kirkcudbrightshire,
Mull of Galloway Granite ......
Another variety from same area .....
A Gneissose Granite from Criffel .....
Granite or QuartzDiorite from head of Loch Doon,
South of Scotland,
Biotite Granite, Loch Dee, South of Scotland
A Diorite identical with that of a dyke near Gutchen
Isle, Colvend shore, south of Dalbeatie,
Other Diorites from the Galloway area ....
Hornblendeporphyrite identical with one found south
of Castle Douglas, Kirkcudbrightshire,
Other Hornblendeporphyrites of the Galloway country .
Hornblendebiotiteporphyrite, "Wigtownshire .
Silurian grits, SouthWest of Scotland ....
Muscovitegranite, Foxdale, Isle of Man
Andesites, Rhyolites, and altered Tuffs of the Borrowdale
series,
Reddish Quartzporphyry, probably from Cushendale,
Antrim,
Reddish granophyres and microgranites, mostly North
East Ireland, but some possibly from West of
Scotland,
A gneissose Grit — locality unknown ....
Carboniferous Limestone ......
Gannister .........
A Muscovitegranite, with microcline and some biotite —
locality unknown.
Pwll Gwaelod.
Gwbert (near Cardigan).
Pwll Gwaelod.
Pwll Gwaelod.
Abermawr, Gwbert (near Cardigan).
Abermawr, Whitesand Bay.
Abermawr (frequent), Aberbach (near
Dinas), Abereiddy.
Pwll Landdu.
Pwll Gwaelod, Aberbach (near Dinas),
Abermawr (frequent), Whitesand
Bay.
Pwll Gwaelod.
Abermawr.
Gwbert (near Cardigan).
Abermawr, Aberbach (near Dinas).
PorthyRhaw, Abermawr, Whitesand
Bay.
Pwll Landdu, Gwbert (near Cardigan),
Aberbach (near Dinas).
Porth Sele
Gwbert (near Cardigan).
Abermawr.
Abermawr.
82 DR T. J. JEHU ON
The most striking fact in connection with the erratics is that so many of them can
be traced to the southwest of Scotland. The Ailsa Craig paisanite has been obtained
in the boulderclay, and is frequently met with on some of the beaches, especially at
Abermawr. The granites, diorites, and porphyrites of the Galloway country are also
well rejDresented, boulders being found which represent the three principal massifs,
namely, (l) Dalbeattie and CrifFel, (2) Cairns Muir of Fleet and New Galloway, and
(3) Loch Doon and Loch Dee, and in addition some from smaller exposures, such as
that of the Mull of Galloway.
The other region from which the boulders have travelled is the northeast of
Ireland, and its rocks are represented in Pembrokeshire by reddish granophyres, quartz
porphyries, and microgranites.
A few boulders are found also which have almost certainly come ultimately from
the Western Isles of Scotland.
It is a noticeable feature that the Lake District rocks are but poorly represented,
and the same is apparently true of those of North Wales.
Many of the boulders and pebbles, such as those of Carboniferous Limestone and the
chalkflints, may have been torn up from the bed of the Irish Sea.
VI. General Conclusions.
The facts adduced in this paper prove conclusively that northern Pembroke
shire has been the theatre of glacial action to an extent greater than had previously
been supposed. Glacial deposits cover the ground in that part much in the same
way as they do further north, and present very similar characteristics. Here also we
meet with a tripartite division of the deposits, namely, a Lower BoulderClay, Inter
mediate Sands and Gravels, and an Upper BoulderClay and RubbleDrift, reminding
us of the tripartite division found at so many places further north on both the east and
west sides of England, and in North Wales. But in the present state of our knowledge
it is very difficult to correlate the deposits found in one area with those found in
another area ; and it is not safe to assume that the Sands and Gravels always represent
any definite horizon in the glacial series.
Of the deposits which have been described it is the Lower BoulderClay which has
the widest extension ; it covers much of the lower grounds inland, and is often hidden
under the other accumulations. It follows the slope of the ground, and a little below
Pen Creigiau Cemmaes it attains an elevation of nearly 600 feet above sealevel. The
series of sands and gravels is a very variable one. Often they taper or die away
suddenly into a stony or loamy drift, and at places are absent altogether. They attain
their greatest elevation near the east end of the area at Pen Creigiau, where the sands
reach a level of over 640 feet, and are followed immediately below by coarse shingly
gravel. The Upper BoulderClay, where the sands and gravels are absent, is some
times seen to rest immediately upon and coalesce with the Lower BoulderClay, so that
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 83
it becomes somewhat difficult to separate them. But the distribution of the Upper
Drift is not so wide ; it is only met with here and there, and the true Upper
BoulderClay is often replaced laterally by BubblyDrift. The more sporadic occurrence
of the Upper Boulder Clay is probably due in part to the fact that it has suffered
more from denudation.
The Lower BoulderClay is undoubtedly the product of an icesheet, and it has
all the characteristics of a true groundmoraine. It is remarkably tough and homo
geneous, and shows no traces of stratification, and it has all the appearance of having
undergone great compression. The included stones are often intensely glaciated, and
are subangular rather than rounded in form. The fact that fragments of marine shells
occur in the clay proves that the ice which gave rise to it must have travelled over
a seabottom. On the other hand, the bits of woody matter sometimes seen embedded
in the tough clay to a depth of 18 or 20 feet suggest that vegetation grew on the
land bordering the sea, before the advent of the ice or during an interglacial period,
and that some fragments of this found their way, by means of streams or otherwise,
to the seabottom, where they lay in the path of the ice. Or they may have been
derived from the remains of a submerged forest. The included erratic stones help
us to follow the direction from which the ice came, and the occurrence of boulders
from the southwest of Scotland and from the northeast of Ireland in the Lower
BoulderClay and Drift as far east as Cardigan, and the discovery of fragments of
marine shells in the Lower BoulderClay exposed at the brickworks near that town,
make it clear that the whole of northern Pembrokeshire was buried underneath
an icesheet coming from the north. The view held by Carvill Lewis, that the Irish
Sea glacier (as he termed it) extended no further south than the extremity of Lleyn
in Carnarvonshire, is shown to be inaccurate. And though Professor James Geikie
makes the mer de glace which overwhelmed Anglesea flow down St George's Channel,
to a limit reaching beyond the southwest of Wales, he only indicates it as crossing
the extreme west of Pembrokeshire at St David's Head. But the facts just mentioned
show that this mer de glace must have passed over a great deal more of Pembrokeshire
than St David's Head. It invaded northern Pembrokeshire along its whole extent, and
even encroached on Cardiganshire to the east, and its trail is evident in the tough
darkblue homogeneous boulderclay, with its northern erratics and the broken shells
derived from the seabottom over which the ice travelled. How much further south
this typical boulderclay or groundmoraine extends is a point which must be left
to future investigation.
This mer de glace was of course the southward extension of that ice sheet which
filled the northern basin of the Irish Sea, and which has been described by Professor
James Geikie and other workers in Glacial Geology. The latest results published
are those of the investigations of Mr Lamplugh in the Isle of Man, and these have
appeared in his Survey Memoir on the Geology of that Island. His observations on
the Irish Sea Glacier are of great interest and importance, and throw light even on
84 DR T. J. JEHU ON
what occurred to the south of his area. Speaking of the conditions which obtained
in the northern part of the Irish Sea at the beginning of the Glacial period he says,
" Along the shores an icefoot probably formed in the winter and broke away in the
summer into floes, which distributed their burden of rockfragments broadcast over
the seafloor. This seems to be the explanation of the universally wide dispersal of
the fragments from Ailsa Craig, which have been recognised in the drift almost all
round the northern part of the Irish Sea basin, in Ireland and Wales, as well as in
the Isle of Man. The seagirt precipices of splintering rock in Ailsa would not fail
to cast off a load upon an icefoot below ; and thus these fragments became strewn
over the seafloor almost as widely as the shells, and were subsequently carried by
the icesheet into nearly every district to the southward where the shells were carried "
(p. 370). This helps to explain also in a satisfactory way the occurrence of fragments
from Ailsa Craig and of boulders from the northeast of Ireland, from the southwest
of Scotland, and even possibly from the Inner Hebrides, in the drift, and on the beaches
of northern Pembrokeshire. For the icesheet as it advanced would pick up any such
fragments which had been previously strewn over the seabottom by icefloes, and
would carry them southwards on to the land as it carried the shellfragments.
Lamplugh estimates that in the neighbourhood of the Isle of Man the ice sheet,
at its maximum, must have attained an elevation of not less than 2000 to 3000 feet
above the present level of the sea, and the general direction of the ice movement
was from northnorthwest to southsoutheast. He points out that " the WestBritish
Icesheet probably attained its ultimate dimensions mainly from the accretion of
snowfall upon its surface, and in only a minor degree from the inflow of tributary
glaciers." He calls attention to certain results which help us to understand the south
ward extension of the icesheet far enough to overwhelm Pembrokeshire. " As the
British icesheets must always have received their increment principally from the
moist Atlantic winds, it seems probable that, without any change of climate, the centre
of greatest accumulation, and consequently of maximum glaciation, would tend to
shift steadily westward and southwestward as the icy plateau rose higher in the
path of the moistureladen winds and compelled their earlier precipitation. This
effect would, moreover, be accentuated by the obliteration of the open water in the
seabasins to the eastward. The WestBritish sheet might from this cause go on
increasing, while its EastBritish and Pennine equivalents were already diminishing
from lack of sufficient snowfall. . . . For the above reason, the shrinkage of the
icesheet covering the Isle of Man is likely to have commenced while the Welsh and
Ivernian sheets were still increasing." Although the ice in the southern part of the
Irish Sea basin did not probably attain such a great thickness as the ice in the
northern area did, nevertheless all the evidence goes to show that even as far south
as Pembrokeshire it must have reached a considerable elevation. The presence of
drift material at Pen Creigiau, at an altitude of over 600 feet above sealevel, indicates
that the icesheet here was in all probability not much less than 1000 feet in thick
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 85
ness, even if we allow that the land at that time stood at a somewhat lower level
than it does at the present day. Mr Lamplugh gives a sketchmap of the Irish
Sea (as far south as the Lleyn promontory), showing glacial strise and probable direction
of the icemovement. The ice which streamed over the Isle of Man from the north
is shown as usual to have travelled south and to have overwhelmed Anglesea, being
here diverted so as to move more to the southwest on account of the opposition of
the ice coming down to meet it from the mountains of Snowdonia. On the western
side of the Irish Sea basin the strise indicate that the ice moved from the land on
the eastern seaboard of Ireland, and took a course from northnorthwest to south
southeast, and coalesced with that which passed over Anglesea and Lleyn. Its course
southwards from this limit is not shown. But in Professor Geikie's map the Irish
ice is made to bend back to the south as a result of its meeting with that part
of the icesheet which flowed over Anglesea, and the northern ice is shown as passing
down to the west of Cardigan Bay, on account of the presence of the ice flowing west
from Merionethshire and Central Wales. But the investigations carried out on the
glaciation of Pembrokeshire make it clear that the Irish ice was not bent back so
sharply, but, on the contrary, it continued in its original course from northnorthwest
to southsoutheast, whilst the ice from the north was forced to invade Cardigan Bay,
and must therefore have shouldered in the Welsh ice again upon the mainland. This is
proved by the direction of the striae seen along the coast, as well as by the presence
of boulders of igneous rock from Ireland and the south of Scotland in the drift as
far east as Cardigan. Again the presence of chalkflint throughout the area is evidence
in the same direction, for these must have come from the northeast of Ireland or
from the bed of the Irish Sea ; and it is possibly from this bed that the boulders of
Carboniferous Limestone which are seen so abundantly at Cardigan have come.
In this connection it is interesting to recall the presence of fragments of Millstone
Grit in the gravels at Pen Creigiau. Our knowledge of the glaciation of Ireland is
as yet very imperfect, and it is difficult to estimate what volume of ice passed seawards
from its eastern border. At the present day the rainfall over Ireland is very excessive,
and so it seems probable that the snowfall was likewise excessive during glacial times.
This would give rise to a proportionately large icesheet moving outwards in all
directions, and so it is quite possible that the amount of ice which found its way into
the Irish Sea basin was considerably greater than has been generally supposed. And
in the southern part of the basin it would to some extent oppose the passage of the
western ice which overflowed Anglesea and the end of Lleyn in a southwesterly
direction, and cause it to turn a little more to the south so as to travel over Cardigan
Bay. The confluent sheet, forming by the junction of the Northern ice, the Irish
ice, and to some extent the Welsh ice, would invade northern Pembrokeshire in the
direction which is shown by the striae along the coast near St David's, namely, from
northwest to southeast, or perhaps from northnorthwest to southsoutheast. This
would explain the transport of boulders from the St David's Head gabbro south
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 14
S6 DE T. J. JEHU ON
eastwards to the neighbourhood of Caerbwdi Bay. In this connection it may be
mentioned that Mr J. Harris, in a report on erratics in South Wales, which appeared
in the British Association Reports, 1898, refers to some boulders found at Pencoed,
near Bridgeend, Glamorganshire. Microscopic sections of some of these were prepared
and sent to petrologists for examination. The result of this is given as follows : — '
" One was identified with the gabbro of St David's Head ; a felsite bore some
resemblance to the preCambrian rocks of Pembrokeshire ; two or three acid rocks,
brecciated felsites, and tuffs are very like those of the Lleyn promontory." From
these data it is concluded that the transport of boulders was from the west or
northwest. If one of the boulders found near Bridgeend is accurately identified as
belonging to the St David's Head gabbro, it is a most remarkable fact. It is
hardly safe to draw any conclusion until it has some further corroboration.
The more or less loose materials covering the bottom of the sea, which existed
before the advance of the ice, would become incorporated into the lower layers of the
icesheet. And as the ice was very thick, and moved onward slowly, it would exert
a great pressure over its bed, with the result that much of the rocky floor would be
torn away, and much of the material ground up and pulverised to form the typical
ground moraine. The shell  banks which occurred on the sea  bottom would be
destroyed, and the marine detritus would be carried forward under the ice or in the
ice. This accounts for the presence of shellfragments at places in the Lower
Boulder Clay. And the most natural explanation of the shelly sands and gravels
is that they represent the material of a seabottom, carried onwards and upwards to
their present position by an icesheet, and rearranged by fluvioglacial action.
That is to say, they are remanies derived from the bottommoraine of an icesheet
which had travelled over a seafloor. Similar sands and gravels have been found
at many other places on the west side of the island, and they have given rise to
much discussion — notably those found at Moel Tryfan in Carnarvonshire. The Pem
brokeshire series differ from those of Moel Tryfan in that they are found overlying
the wellmarked stiff Lower BoulderClay. The most remarkable feature in connection
with the Moel Tryfan beds is the great elevation at which they are found — 1350 feet
above sealevel. In Pembrokeshire the greatest height at which they have been met
with is 642 feet at Pen Creigiau, four miles southwest of Cardigan. The mode of
origin of such sands and gravels has been one of the most vexed questions in
Glacial Geology. Some writers, such as Mackintosh, T. M. Reade, and others, have
argued that the sands and gravels represent marine deposits laid down in place
during a great submergence. It is admitted, even by the opponents of that theory,
that a partial submergence took place during Glacial times, but to what extent is
not known, and there is no evidence to show that it meant a sinking of the land
in Carnarvonshire to as much as 1350 feet below its present level. And the partial
subsidence which is allowed is generally thought to have diminished towards the south.
In Pembrokeshire no evidence can be seen along the coast which would lead us to
THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 87
believe that there had been a subsidence of 640 feet below the present level, so as to
account for the marine deposition of the beds found at Pen Creigiau. Further, if the
advocates of a submergence point to the presence of marine shells at places in the
sands and gravels as a proof, on the ground that the shells must have been altogether
destroyed if carried beneath the ice in the morainic debris, one asks then how they can
account for the presence of fragments of shells — some large enough to be identified —
in the tough blue boulder clay underlying the sands and gravels ? No one who looks
upon that clay, as exposed, for instance, at the Cardigan brickworks, can doubt for a
moment that it is the product of an icesheet. Not a trace of stratification can be seen
in it, nor is there any character which suggests even the possibility of its being the
result of marine deposition. And yet marine shells are seen imbedded in the clay.
This, as has been already pointed out, is due to the fact that the icesheet, of which this
clay is the bottommoraine, travelled over a preexisting seabottom. The fact that
everywhere the shells are very broken and much rolled is hardly compatible with the
view that they are now found in or near the positions in which the molluscs themselves
lived. And it is worthy of note that the Lamellibranch shells obtained in these sands
and gravels are never found with the two valves in apposition, as one might expect to
find if they lie in ordinary sea deposits. Again it has often been pointed out that it is a
significant fact that deposits of this kind only occur in glaciated areas, and that wher
ever broken shells are found, with them there also we find fartravelled erratics present.
And this is to a marked extent the case in Pembrokeshire.
Mr J. F. Blake (Geol. Mag., vol. x., 1893, p. 267) concluded that the shelly sand at
Moel Tryfan had been pushed up in front of the advancing glacier, and that, as a result
of this glacier meeting that which came out of the Bettws Garmon valley, the sand
got pushed into a protected corner and was left there. But in northern Pembrokeshire
the sands and gravels are found scattered in patches over a wide area, and are
frequently well bedded. Here they are the products of the washing and resorting of
infra and intraglacial detritus. This may have gone on partly under the ice, but it
would no doubt take place to a great extent at the time of the melting of the icesheet,
when large streams would issue from the margins of the glacier and rearrange much of
the superficial deposits left on the surface of the land.
The Upper BoulderClay is so sporadic in its occurrence that it is difficult to draw
any definite conclusions with regard to it. It may possibly represent a second advance
of the icesheet after an interval of less severe glacial condibions. It is far more stony
than the Lower BoulderClay, and in places passes into Rubbly Drift. This RubblyDrift
is very similar to that found by Lamplugh in the Isle of Man, and is probably " the
remanie deposit of the icesheet modified by subaerial agencies." At the time of the
final disappearance of the ice, torrential waters must have overflown parts of the surface,
and the rubble is probably to be attributed in part to the action of these waters.
Morainic material would become mixed up with rock debris, formed by ordinary
weathering processes, and the whole mass would be rearranged, and in places sifted
by the waters.
Trans. Roy. Soc. Edin.
Vol. XLI.
2
o
o
2
S3
3
B
H
•3
15
H
C
n
i
»1
1
* >
' 4
:»/•'>
»
•>
03
r;sA i " ;
P3
( 89 )
V. — Spectroscopic Observations of the Rotation of the Sun. By Dr J. Halm,
Assistant Astronomer at the Eoyal Observatory, and Lecturer in Astronomy at
the University, Edinburgh. Communicated by The Astronomer Royal for
Scotland.
(MS. received February 19, 1904. Read March 21, 1904. Issued separately May 4, 1904.)
The causes of the peculiarities of the solar rotation exhibited by the superficial layers
of the sun's body must still be considered unknown, notwithstanding recent interesting
attempts at explanation. This is no doubt partly due to the difficulties of the hydro
dynamical problem placed before the mathematician in an investigation of such complexity
as the movements of particles in a rotating fluid subject to energetic convection. But it
must also be conceded that the observational data available for a basis of mathematical,
or even speculative, research are still so scanty, that for this reason alone we may
perhaps not feel surprised at the failure so far of theoretical attempts.
The first empirical demonstration of the peculiar law which apparently governs the
rotation of our luminary was given by Carrington, whose important investigations were
taken up and extended by Spoerer. Their observations brought to light the main
character of the peculiarity of solar rotation, viz., the decrease of the angular velocity
from the equator towards the poles. An objection has, however, been raised against
the value of their results with regard to the general rotation of the sun's photosphere,
on the ground that they were obtained from the observed movements of solar spots.
It was justly urged that spots have " proper motions " which preclude their adoption as
points of reference. Besides, we must remember that these spots were visible only
within an equatorial zone of about ±50° or 40° latitude, and that therefore the polar
regions remained inaccessible by this method. This difficulty and limitation was over
come by the ingenious application of the spectroscope to the problem, which we owe to
Professor Duner. His conclusions are based on the displacements shown by the
Fraunhofer lines at the solar limb, where the gases producing these absorptions are
carried by the rotation either towards us or from us. His results are, it is true, not
directly comparable to those derived from the movements of the spots, because both refer
most probably to different levels, and therefore perhaps to different conditions of motion ;
but the great advantage of the spectroscopic method seems to me to lie in the fact that
we always measure at the same level, wherever this level may be — a point on which we
are by no means certain in the case of the spots. Besides, we are independent of the
uncontrollable vicissitudes of proper motions, and we are able to extend the investiga
tions from the equator to the immediate vicinity of the poles. The results obtained by
Professor Duner may be summarised by the statement that the retardation of the
angular velocity discovered by Carrington and Spoerer was found to be also shared
by the photospheric layer emitting the Fraunhofer lines, and that the amount of this
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 5). 15
90
DR J. HALM ON
" lag" appeared to increase continually towards the poles. So far, then, as the general
character of this peculiarity goes, the question appears to be empirically settled. But
there remains still another, and, as I think, not less important question. Are we
allowed to suppose that the surface rotation of the sun remains unaffected by the
periodic changes of solar activity ? Judging the question from a purely logical point of
view, we are almost bound to answer it in the negative. It seems to me difficult to
imaoine that such violent disturbances of the normal conditions of convection as we
perceive in solar eruptions and spots, and the consequent displacements of matter in the
solar olobe, should have no influence on the distribution of the rotational velocities at the
surface. A careful study of the behaviour of the solar rotation during a cycle of
activity may probably teach us far more about the causes and the seat of these solar
Spectrum of receding
limb.
Solar limbs.
Spectrum of approach
ing limb.
Fig. 1.
Group of lines as seen in the viewing telescope. (1 and 3 are telluric, 2 and 4 solar line s. )
disturbances than the whole array of statistical facts regarding the periodic displays of
dynamical phenomena at the surface which are now in our possession. Professor
Duner's observations, covering a period of three years, during which next to no change
took place in the activity of the sun, cannot give an answer to this question. I there
fore thought it a promising venture, the success of which seemed to me in some way
guaranteed by the great accuracy and consistency of Duner's results, to extend these
observations over a time of more pronounced changes of solar activity. The following
contains a description of the results so far obtained, and of the instrument employed in
my investigations.
The observations were begun in August 1901, and so far carried on to the end of
1903. Although this interval of time is not greater than that covered by Duner's work,
there is this essential difference, that the year 1903 was characterised by an abrupt and
violent increase of solar disturbances after a pronounced and persistent calm during
19012. This fact, as we shall see, has an important bearing on the results, which
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 91
seem to me so remarkable and unexpected, that I trust this early publication may
induce other observatories to take part in the work, at this opportune time of the
beginning of what seems to be an energetic sunspot cycle.
As I intended to make the following observations directly comparable to those of
Professor Duner, I have used the same group of lines. The wavelengths of these are
given on page 55 of his treatise, " Eecherches sur la rotation du Soleil." # The
measurements therefore refer to the same photospheric level. Fig. 1 represents this
group of lines as seen in the viewing telescope. The displacements of the solar lines
shown in the figure correspond approximately to the shift at the solar equator.
With regard to the instrument, some deviations from Duner's arrangement have
been suggested by the apparatus at my disposal. Since they increase the stability of
the instrumental plant, these alterations may be considered as essential improvements.
The most important of them was attained by the use of a siderostat. Duner's spectro
scope was mounted on the great refractor of the Lund Observatory. The extraordinary
dimensions of the apparatus gave rise to flexures which were bound to lessen the
accuracy of the observations. The influence of such flexures is, of course, avoided if
the whole spectral apparatus, including the front telescope, is mounted on fixed and
insulated tables, and if the solar light is thrown upon the objectglass of the telescope
by means of a siderostat. The considerable advantage of such an arrangement is indeed
shown by the fact that the probable error of a single observation appears to be only
half the probable error of one of Professor Duner's measurements. This increase of
accuracy has to be ascribed chiefly to the greater stability of the instrument, and
perhaps also to the easy and comfortable position of the observer during the
observations.
A second essential departure from the design of the apparatus used by Professor
Duner is to be found in the arrangement by which the focal images of the two opposite
limbs of the sun are thrown upon the slit of the spectroscope. Duner employed a
system of rightangled prisms arranged according to a device previously suggested by
Langley. By successive inner reflections from the hypotenuse surfaces of these prisms
the light of the solar limbs can be thrown upon neighbouring points near the centre of
the slit, which lies in the optical axis of the telescope. In a much simpler way,
however, this same purpose can be attained by using as a front telescope a heliometer
of sufficient optical power. By separating the halves of the objectglass, we can at
once bring opposite points of the solar limb into contact, and these images may be
thrown upon the centre of the slit without any further auxiliary apparatus. Besides,
by altering the position angle of the heliometer, all the opposite points of the solar disc
can be successively brought into contact. Thus we are enabled to determine the
rotational velocity for any desired heliographic latitude simply by turning the
heliometer into a position which corresponds to that latitude.
It seemed advantageous to throw the two solar images at each observation into such
a position that the line joining their centres coincided with the slit. This could be
* Nova Acta Eegiae Societatis Scientiarum Upsalie?isis, 3rd series, vol. xiv. fasc. ii., 1891.
92 DR J HALM ON
effected by bringing a largesized, rightangled prism into the cone of light between the
objectglass of the heliometer and the focus. By turning the prism, the solar images
projected upon the slitplate could be brought into the desired position, which is
represented in the accompanying fig. 2. I may say that in order to attain sufficient
accuracy of this adjustment, two lines parallel to the slit and at equal distances from it
had been engraved upon the slitplate. The solar images were brought into such a
position that the two lines cut off equal segments. This arrangement was quite
sufficient to guarantee the heliographic latitudes of the points measured within a
fraction of a degree. Such small errors in the adjustment have, however, no appreciable
effect on the observations, because they displace the measured point on the one limb
exactly as much towards the equator as they displace the point measured on the other
limb towards the pole. The total displacement therefore still agrees practically with
that we should have obtained if no such error had been present. This consideration,
however, does not apply to points exactly on the equator.
Fig. 2.
The heliometer employed in these observations has been kindly lent to this Observatory
by the Hon. Lord M'Laren, Judge of the High Court of Session of Scotland. It is the
instrument used by Sir David Gill in the Mauritius Expedition, 1874, and a full
description of it may be found in vol. ii. of the Dunecht Observatory publications. Its
aperture is 4*2 inches, and the focal length about 64 inches. The optical quality of the
glass is exceedingly good. For the purpose of the observations the eyeend part of the
heliometer had to be removed, so that the focus came to fall outside the main tube. The
instrument is mounted upon a castiron table, adjustable in two horizontal directions at
right angles, with the optical axis parallel to the line of the meridian, the objectglass
towards the north. As a collimator, a telescope is used of 4 inches aperture and 50'5
inches focal length. The eye end tube, which carries the slit, is adjustable by means of
a focussing screw. The cylinder of rays, emanating from the objectglass of the collima
tor, is thrown upon a Rowland plane grating of speculum metal, 5 inches long, 3^ inches
broad, with 14,438 lines to the inch. The surface of the grating is perpendicular to the
axis of the collimator. I have used the spectrum of the third order, which, upon the
one side of this grating, is remarkably bright and welldefined. The disturbing effect of
the overlapping violet spectrum of the higher order was found to be sufficiently elim
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 93
inated by a plane glass of purple colour fixed in front of the eyepiece. The grating
is mounted upon a stand, turning on a vertical axis in such a way that the axis of
rotation coincides exactly with the plane of the grating. This stand carries a horizontal
circle, divided from 5 to 5 minutes, which can be read by two diametrically opposite
microscopes. By this arrangement a differential determination of wavelengths is made
possible.
The viewing telescope, which is horizontally mounted on another isolated stand, has
an objectglass of 4*1 inches aperture and 60 inches focal length. A Cooke wire microm
eter is used for measuring the distances between the lines. After various preliminary
trials with crossed and parallel wires, the pointing on the spectral lines by means of
sufficiently close parallel wires seemed to me to yield the best results. The observation
consists in bringing, by turning the micrometer screw, the centre of the space between
two close wires exactly upon the line to be measured, in the same way as the meridian
circle observer sets the division of the circle between the wires of the microscope. Care
has to be taken, however, that the setting is made exactly on the solar limb. I have
found that this may be done without the least difficulty. The errors of the micrometer
screw have been repeatedly determined, with the result that the progressive error may
be considered as negligible, but that there are indications of a small periodic error. In
order to eliminate the effect of the latter directly from the observations, I have invariably
observed the lines with two pairs of wires which were at a distance of exactly 1 1 turns
of the screw. As is well known, the periodic error is practically eliminated from the
arithmetical mean of the observations made with two sets of wires thus arranged. The
value of a revolution of the screw, expressed in wavelengths, has been determined by
measurements of the distances of neighbouring spectral lines, by a method which is fully
described in Professor Duner's paper. Such measurements will be continued in the
future, but there can be no doubt already that the provisional value here adopted must
be sufficiently correct, and that therefore the computed values of the velocities may not
have to be altered in the final discussion. The value of the screw is subject to small
variations, chiefly owing to temperature fluctuations. These have, however, been taken
into account in the reductions in the following way : — Each set of observations yields a
very accurate determination of the normal distances between the four lines of the group
measured. Since the wavelengths of these lines are known, the value of a revolution
of the screw expressed in wavelengths may be found by dividing the measured distances
of each pair of lines into the difference of their wavelengths. Thus values of a turn of
the screw are found for different points within the group, and from these the value to
be applied in the reduction can be easily derived.
The computation of the position angles of the various points measured along the
solar limb requires the knowledge of the rotation of the field of the siderostat at the
time of the observation. Obviously, what is required is the diurnal rotation of the hour
circle of the sun, and therefore also of the pole of the heavens, round the centre of the
field. The data required for this computation have been supplied by Cornu in his
94 DR J. HALM ON
paper on the " Law of Diurnal Rotation of the Optical Field of the Siderostat and
Heliostat" {AstrophysicalJournal, vol. xi., 1900, pp. 148162). It appears from his
calculations that in our case, where a siderostat oriented in the meridian has been used,
the angle Y at the centre of the field between the reflected image of the meridian and
that of the hour circle of the sun is expressed by the formula,
tan£Y = K tan£,
where t is the hour angle, reckoned positive towards the west, and
K _ Bin}(0j?)
sin H0+P)'
</> being the latitude of the place and p being the polar distance of the sun. These
equations determine the amount and the direction of the rotation. We are thereby
enabled to fix the position of the north point of the solar disc as it appears in the field
of the siderostat for any time of the day. The values Y can be tabulated with the
arguments t and declination of the sun. The position circle of the heliometer is so
oriented that the diameter 0°180° falls in the plane of the meridian. The angular
distance of the point under observation from the north point of the solar disc, that is,
the position angle P of the point, is then found by the expression
P=90°n+Y,
where II is the reading of the heliometer position circle. The value of P being found,
the heliographic latitude of the point observed can be directly computed from the
formulae given by Professor Duner in his treatise, where he also exhibits extensive
tables, greatly facilitating these computations. I need not, therefore, enter more fully
upon this part of the reductions.
I shall now briefly describe the way in which observations were performed with this
instrument. The sunlight reflected from the siderostat mounted on the main platform
of the Observatory is thrown into a meridional direction upon a window of 6 inches
aperture in the north wall of the great optical room, and falls upon the objectglass of
the heliometer placed immediately behind this window. The halves of the objectglass
are screwed apart until the two solar images are nearly in contact, but still separated by
a narrow space, which in the viewing telescope appears as a dark horizontal band between
the spectra of the two limbs. The rightangled prism mentioned above is then turned
and the heliometer adjusted so that the solar images fall upon the slit in the position
indicated in fig. 2. After reading the position circle of the heliometer, the observer
commences the measurements by pointing the first pair of the micrometer wires on the
four lines of the upper of the two spectra seen in the viewing telescope. He begins,
say, with the lefthand line of the group, and proceeds towards the right. He then
measures with the same pair the lines of the lower of the two spectra in the direction
from right to left. The observations are afterwards repeated with the second pair of wires ;
this time, however, they are begun on the lower spectrum from left to right, and finally
the upper spectrum is measured from right to left. This arrangement, though it may
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 95
seem perhaps somewhat pedantic, has certainly the advantage of referring the mean of
the observations on each limb to one and the same moment of time. Besides, approach
ing a line alternately in opposite directions must decrease the amount of the personal
error of the observer in his judgment of the bisection. After the completion of these
measurements, the halves of the heliometer objectglass are reversed and the time noted.
The observations are then repeated in the same order as before. A complete set there
fore consists of 32 pointings, 1G on each limb. When the set is finished, the heliometer
is turned 10° in position angle, and the solar images are again brought into the position
of fig. 2. The observation of a full set, including the necessary adjustments of the
instrument, requires on the average from 10 to 15 minutes.
The observations of Professor Duner refer to six equidistant points on the solar
limb between the equator and 75° latitude. The selection of these fixed heliographic
latitudes was perhaps necessitated by the arrangement of his instrument, which
required a previous computation of the difference of declination between the north
limb of the sun and the point to be observed. Since, however, a previous knowledge
of the heliographic latitude is not required in our case, I have preferred to proceed
from 10 to 10 degrees on the position circle of the heliometer, without considering
at all the heliographic position of the points thrown upon the slit. The readings of
the position circle, together with the time of the observation, are sufficient to evaluate
afterwards the true heliographic latitude. In this way, a uniform and continuous
distribution of observations over the whole quadrant, from the equator to the pole, may
be secured, and we are, I think, in a better position to ascertain the character of the
velocity curve by this method than by the observations of six single points of the
quadrant.
After this general description of the instrument and the method of observation, I
shall now turn to the results. The measurements were commenced on 13th August
1901 and extended to 6th November 1903. During this time 564 determinations of
the rotational velocity were made. The values obtained, expressed in kilometres per
second, were divided into two groups ; the first group comprising the measurements
made during 19012, the second those of the year 1903. The reason for this division
is that the period 19012 was characterised by a low and protracted minimum of solar
activity, while in the early part of 1903 the commencement of a new solar cycle was
vigorously manifested by the appearance of large spots. It was therefore to be expected
that, if indeed solar activity has an influence on the rotation of the sun, such a division
into groups as I have made would show this influence more clearly than any other. In
each group the individual values were arranged according to their heliographic latitudes,
and from the materials thus collected normal values were formed by adequate combina
tions of single values into arithmetic means. The figures thus obtained are exhibited
in the following table.
96
DR J. HALM ON
Table I.
19012.
1903.
Heliographic
Linear Velocity
No. of
Heliographic
Linear Velocity
No. of
Latitude.
per sec.
Obs.
Latitude.
per sec.
Obs.
km.
km.
16°
1908
33
37°
1898
16
5*2
1894
30
91
1883
15
83
1871
30
155
1831
16
128
1802
30
211
1753
15
180
1720
30
277
1631
16
238
1594
30
340
1512
16
306
1488
30
403
1365
15
387
1265
30
471
1201
16
473
1061
30
547
1014
16
558
0840
30
623
0797
15
651
0560
30
692
0564
16
755
0307
30
760
0408
15
830
0187
14
The values of the linear velocities were now plotted upon squared paper as ordinates,
with the heliographic latitudes as abscissae, and in each group a curve was drawn repre
senting, as closely as possible, the observed ordinates. In this graphical form the results
are represented in the accompanying fig. 3. It is seen at first glance that the two
groups differ materially from each other. They agree fairly well only at the equator,
but in higher latitudes the observations of 1903 show considerably larger values than
those of the two preceding years. There is not a single exception to this rule. The
evidence of these curves leaves therefore no doubt as to the reality of this systematic
difference, which is also apparent from a consideration of the probable error. An in
vestigation of the observations of 1903 shows the probable error of a single observation
to be ±0*070 km., a value which appears to be very nearly constant for all helio
graphic latitudes. From this we find the probable error of a mean value for the first
group ±0*013 km., and for the second group ±0*018 km., while the observed
systematic differences between the groups attain the considerable value 0*16 km. in
middle latitudes.
Professor Duner has also computed the probable error of his observations. I find
from his figures that of a single observation to be ±0*138 km., also nearly constant for
all latitudes. Considering that his single observations comprise twelve to twentyfour
pointings of each line, as compared with only eight in my observations, we may conclude
that the latter show at least double the accuracy of those of Professor Duner. As
already pointed out, this favourable result is in my opinion only due to the greater
stability and the more comfortable management of the instrumental plant.
The influence of a possible error in the position angles shown by the heliometer has
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN.
97
also been closely investigated. The careful mounting and orientation of the instruments,
it is true, did not warrant the assumption that such an error was to be seriously appre
hended. Besides, the arrangement of the observations was such that an error of this
kind, even though it existed, should practically disappear from the mean values. This
GM^O. \~AMTVLd jnxrhM/nxi/ Vnxi AxnnAxyrf rotaZu^ruxL vJuxuXajla oZ^vrouL
2. titti
JajurxmA, AhjU Axmojr* 190102 a/rv<L 1^)03 '
19
18
^x^
*
^J>Q
1.T
1.6
\
1.5
\
\
14
s \
1.3
\
\
\
1.2
*
» \
1.1
1.0
\
\
09
0.8
'
\ \
\ \
07
\
\
0.6
\ \
\ \
0.5
\
0.4
\
\
, N^.
0.3
\ \
\ \
0.2
T \
\
\
01
\
\ \
N. \
o.o
S \
\\
ts:)°
10°
20°
3o*
4o°
5o c
60'
70"
80
90
may be seen from the following consideration. Let the error of the position angle be
Ap, so that a point on the solar limb whose computed distance from the equator,
measured in the direction of the position angle, is II, is actually at a heliographic latitude
II + Ap. If, now, we pass to the opposite side of the pole to a point whose computed
distance from the equator, again measured in the direction of the position angle, is
180° — IT, such a point would actually be in the heliographic latitude II — Ap. Hence
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 5). ] 6
98
DR J. HALM ON
our first point would be nearer the pole and the second nearer the equator than is
shown by computation, and the arithmetic mean of both must therefore agree with the
actual latitude. If, then, we arrange, as has indeed been done in these observations, the
measurements so that equal numbers of points are observed on opposite sides of the
poles, the mean values of the heliographic latitudes are practically uninfluenced by a
systematic error of the position ang]e. The question, however, whether such an error
actually exists, is readily answered by the observations. We have only to separate the
QJUX^t Micywruj/ Mvo uada) V Ate u iw Mw Crvo jru^iodi ly&l'Z
«m,A 1905
'■9
ts
17
1.6
1.5
14
13
1.2
"*.__
V
10°
20°
3 0°
40°
SO"
60°
70'
80°
9 0°
observations into two groups in such a manner that the heliographic latitudes increase
with the position angle in one group, and decrease in the other. The error would thus
show itself by the fact that the same values of the velocity would not correspond to
exactly the same latitudes in the two groups ; or, to put it in other words, that corre
sponding latitudes should show slightly different values of the velocities. I have made
such investigation for the whole time from 1901 to 1903, with the following result : —
Table II.
Heliographic
Group I.
Group II.
Latitude.
km.
km.
825°
021
022
750
034
041
650
065
064
550
092
091
450
118
117
350
143
142
250
L64
163
150
177
179
50
189
189
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 99
With the exception of latitude 75°, the differences between the two groups are far
within the limits of their probable errors. This shows, as was anticipated, that practically
no error of the position angle was present.
We have now to deal with the question, How are these remarkable differences
between the two groups to be interpreted ? Is their cause indeed to be sought for in
the sun itself? We are, of course, not yet in a position to attempt an answer to this
question, but there is a probability of the solar origin of this phenomenon, for which I
should like to give some reasons. First, we have to take into consideration that the
observations have been made throughout by the same observer, with the same entirely
unaltered instrument, and at the same place, — that means, under the same atmospheric
conditions. While it must be granted that deficiencies in the instrument and in the
method of observation, or the personal perception of the linedisplacements, which may
be different for different observers, but most of all atmospheric conditions, which shall
be discussed later — that all these circumstances may vitiate the results, there seems,
however, to be no reason for the assumption that these influences should have altered
so decidedly from one year to the next.
Secondly, that a real change of the rotation must have taken place from the one
group to the other seems to be indicated by a significant peculiarity of the angular
velocities which have been derived from the linear velocities v contained in Table I.
By multiplying the values of v by sec. /3, /3 being the heliographic latitude, we obtain
numbers which obviously must be proportional to the angular velocities. This computa
tion having been made, the values were plotted down in fig. 4, again for both groups
separately. At first glance we recognise the retardation of the angular velocity from
the equator towards the poles, but it will be remarked that the amount of this retarda
tion during the second period is considerably smaller than during the first. The most
significant fact, however, appears to be this : if in both groups the values of the curve
for corresponding latitudes are subtracted from the equatorial velocity, these differences
can be made to agree perfectly if the values for 19012 are multiplied by the factor 0'4.
This result seemed to me so remarkable that I decided to test its correctness by a special
investigation, including also the results of Professor Duner's observations.
First of all I endeavoured to find an empirical formula which should represent the
angular velocities in every group in a satisfactory manner. After various attempts at
representing these velocities in the usual way by the sine and cosine functions of the
heliographic latitude, the method had to be abandoned, as my observations could be
represented only by extremely complicated expressions of such a form. Accidentally,
however, I arrived at a formula very different from those hitherto used, which satisfies
the observations in all three groups with a high degree of accuracy, and which has the
further advantage of being extremely simple. This formula can be expressed in the
following way —
£ = a — bc $ ,
where £ is the angular velocity in latitude /3, and a, b and c are constants. When
100
DR J. HALM ON
computing the constants of this entirely empirical formula, I found that c had the same
value for all the three groups, but that a and b showed considerable differences. The
value of c was found to be 1'01447 if /3 is expressed in degrees, and 2'2784 if
expressed in units of the sun's radius. The constants a and b in each of the three
groups are given in the following table, which also contains the comparison between the
observed values of v and those computed by means of the preceding formula.
Table III.
Group
a
b
18879 (Duner)
2349
0354
19012 (Halm)
1903 do.
2292
2066
0370
0148
Group
18879.
04°
v (comp.)
199 km.
v (observed)
198 km.
obs. — comp.
ooi
150
185
185
00
300
156
158
+ 002
450
119
119
ooo
600
076
074
002
748
034
Group
034
19012.
ooo
16°
v (comp.)
1908
v (observed)
1908
obs.  comp
oooo
52
1881
1894
+ 0013
83
1851
1871
+ 0020
128
1797
1802
+ 0005
180
1720
1720
0000
238
1617
1594
0023
306
1474
1488
+ 0014
387
1281
1265
0016
473
1055
1061
+ 0006
558
0822
0840
+ 0018
651
0566
0560
0006
755
0299
Groui
0307
1903.
+ 0008
P
37°
v (comp.)
1906
v (observed)
1898
obs.  comp.
0008
91
1874
1883
+ 0009
15 "5
1813
1831
+ 0018
211
1741
1753
+ 0012
277
1633
1631
0002
340
1513
1512
oooi
403
1374
1365
0009
471
1209
1207
0008
547
1008
1014
+ 0006
623
0792
0797
+ 0005
692
0592
0564
0028
760
0393
0408
+ 0015
830
0192
0187
0005
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 101
The probable value of a difference " obs. — comp." is in all the groups approximately
the same, viz., ±0  010 km., whereas the probable errors of an observed single value of v
are respectively ±0*013, 0'013, 0"018. The formula therefore represents the obser
vations in all the groups with sufficient accuracy. The constancy of c is a somewhat
significant feature. If it should be confirmed by future observations, it would almost
certainly point to the conclusion that the formula is not purely empirical, but has some
physical meaning. A decision on this question would be highly important, for if the
rotation of the surface layers be actually governed by a law which has its analytical
expression in the formula here adopted, we may conclude that surface phenomena are
the cause of these peculiarities. I shall not dwell, however, on this at present, but
shall now draw attention to the result of these computations with regard to the
constants a and b of our equation. Here we find a satisfactory agreement between the
values for 18879 and 19012. These are also the two groups which have exactly
the same position in the sunspot cycle, both embracing a time of minimum activity.
On the other hand, as already pointed out, the year 1903 was characterised by
vigorous displays of spots and solar eruptions. Simultaneously with this activity we
notice an enormous change in the values of the constants of our equation. The
" retardation " of the higher latitudes appears now to be reduced to less than half its
former amount. This is a novel and, I am sure, unexpected result. So far, it is true,
it can only be said to represent a coincidence in time, but I trust the discussion has
made it sufficiently clear that the discrepancies observed, since they can in no way be
ascribed to observational errors, must, in all probability, be interpreted as a new and
very peculiar feature of the still mysterious mechanism of the sun. Care has been
taken throughout not only to investigate all possible errors which may arise from
deficiencies of the instrument, and to eliminate such errors from the reduced values
of the observations, but the measurements were also arranged in such a way that
the possible remainders of such errors which could not be elicited in the reductions
should have a minimum effect on the results.
The possible bearing of an investigation of this kind on important questions of solar
physics makes it very desirable that the observations should not only be continued at
this Observatory, but that other observatories should also take part in them. Next to
the question of possible alterations in the amount of solar heat, an answer to which may
now be expected from the ingenious researches of Professor Langley and his staff, the
problem of solar rotation should command the greatest attention from the part of
solar physicists, for no other seems to me so well adapted to give us information on the
mechanism of the solar forces. That the periodic play of these forces should in some
manner be conducive to changes of the distribution of the moments of rotation is a
logical conclusion on which I trust astronomers are unanimous. It is well to remark in
this connection that the idea has already received theoretical consideration in an in
teresting paper by Mr Emden.
The construction of the apparatus required for these observations is well within the
102
DR J. HALM ON
reach of our large astrophysical observatories, and there are several beautiful helio
meters which might be resuscitated for this purpose from their present unprofitable state
of repose. Nor is the present the only investigation which can be taken up with an
instrument of the kind here described. The construction of the apparatus, combined
with its higli dispersive power, makes it possible to separate by a mere glance the
telluric lines from the solar ones, and at the same time to determine by a single
observation the wavelength of a line with an accuracy of about ±0'05 tenthmetres.
The instrument appears therefore to be particularly adapted for an investigation of the
telluric spectrum in accordance with the method first suggested by Cornu.
As regards the accuracy of the determination of the rotational velocities, I am
confident that, under more favourable atmospheric conditions, the annual output of
observations can be considerably increased, and thereby the probable error of the
annual means correspondingly lessened. It does not seem to me impossible that in this
way changes of only one to two hundredths of a kilometre could be traced with certainty.
I must not conclude this paper, however, without drawing attention to an incon
venience encountered in this spectroscopic determination of the rotational velocity.
AVith a hazy atmosphere, a not inconsiderable quantity of scattered daylight, reflected
from the particles of the aqueous vapour contained in the air, is thrown upon the slit of
the spectroscope. As a certain percentage of this light will have emanated from
the interior of the sun's disc, the solar lines of this " day " spectrum appear less dis
placed than the lines of the two solar limbs. By a superposition of the two spectra,
the intensity of the lines will therefore be lessened on the one side and increased on the
other, the effect always being to bring the lines nearer to their normal position. The
velocities obtained under such conditions must therefore be too small. I have often had
occasion to observe this phenomenon when, after a measurement made under favourable
atmospheric conditions, the sky was suddenly overspread with a veil of haze, a meteoro
logical feature not at all infrequent in the climate of Edinburgh. When the observation
was then repeated, the measured displacement was invariably found less than that
obtained before. Observations under misty conditions of the sky were therefore care
fully avoided. A reliable scale for the transparency of the atmosphere was supplied by
the optical appearance of the dark band separating the spectra of the two limbs in the
viewing telescope. Whenever this space became so bright as to show traces of the
absorption lines of the solar spectrum, the observation was broken off. All the measure
ments used in this discussion were made at moments when the band mentioned was
uniformly dark, so that the spectrum of the diffused daylight cannot have seriously
influenced the measured displacements.
SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN.
103
Table IV.
Single Values of Linear Rotational Velocities of the Sun observed
during the Years 19011903.
Lat. Vel.
1901.
Aug. 13.
242 141
381
504
641
770
792
646
485
329
170
Aug.
230
Aug.
85
56
198
333
476
607
740
Aug.
115
2*5
167
308
445
583
718
779
650
510
371
236
94
45
Aug.
10
22
114
252
390
528
665
45
55
211
Aug.
23
123
084
055
022
012
071
097
117
149
15.
152
16.
2
1
1
1
1
19
1
1
1
1
1
1
1
1
1
1
20
2
1
1
1
1
1
1
1
03
77
76
45
17
67
34
70
85
60
23
20
57
16
33
51
08
28
43
48
75
03
80
59
39
08
84
32
93
80
52
21.
179
Lat.
34
55
149
237
23
314
404
493
584
672
759
55
105
Vel.
•69
•75
•65
•58
•75
•31
•13
•83
•66
•35
•32
•77
•83
Aug.
22.
08
21
82
1
170
1
255
r
343
r
432
i
522
i
6M
o
697
o
54
2
66
2
•07
•87
•84
•83
•46
•19
•03
•75
•42
06
•11
Aug
37
50
34
111
205
291
349
442
Aug.
786
688
581
474
367
258
146
38
72
79
267
178
24.
•06
•11
•96
•82
•67
•54
•20
•08
25.
011
052
089
119
141
164
187
198
205
204
184
186
Aug. 27.
66 200
81 185
Lat.
05
169
Vel.
2 05
209
Aug.
28.
100
189
186
192
274
183
363
159
435
133
524
107
510
072
11
190
Aug.
30.
193
170
25
188
83
189
Aug.
31.
19
217
11
195
Sept
1.
65
190
193
197
103
203
18
213
70
2 09
159
198
Sept.
506
592
680
22
81
10
95
183
270
2.
103
076
032
194
188
217
192
183
159
Sept. 4.
288 i53
174 171
57 203
59 221
Sept. 5.
08 2 02
Sept. 7.
77 185
411 134
489 119
Lat. Vel.
Sept. 10.
629 077
522
414
308
197
87
Sept.
159
244
333
424
51 3
600
686
Oct.
69
18
103
189
Nov.
396
482
563
646
732
12
90
168
104
120
159
187
192
11.
199
177
167
137
100
087
056
26.
198
211
184
149
5.
130
099
086
063
039
200
173
167
1902.
Feb. 12.
130 172
53
27
105
189
189
201
173
168
Feb. 13.
79 214
Feb. 15.
33 185
146
261
372
489
607
71'4
179
146
107
098
077
042
Lat. Vel.
Feb.
678
550
Feb.
357
271
189
103
18
68
148
23
315
398
Mar.
460
545
630
7L4
Mar.
762
659
547
433
323
139
2'9
Mar.
210
92
18
267
355
439
515
17.
031
063
18.
139
135
146
1 71
167
174
L75
167
152
114
7.
096
078
051
035
15.
023
055
088
125
143
190
205
16.
164
190
189
L47
122
104
084
Mar. 20.
95 177
68 184
181 1'76
Mar. 21.
176 166
Mar.
65
02
76
160
243
22.
185
177
190
183
166
Lat.
Vel.
321
154
407
132
492
099
Mar.
24.
620
064
Mar.
25.
706
044
598
075
488
110
378
128
264
161
155
190
42
196
73
185
Mar.
29.
211
158
Apr
67
. 1.
189
Apr
04
2
177
84
189
165
175
256
150
314
148
401
114
Apr
501
.7.
098
Apr.
526
10.
104
615
086
706
067
785
016
Apr.
795
16.
024
711
046
759
013
655
061
549
085
443
114
336
146
2L8
155
108
163
0'4
183
Apr.
51
18.
185
160
180
Lat.
Vel.
Apr. 25.
180 161
273 173
Apr.
279
171
64
50
161
275
Apr.
181
28.
172
166
184
174
147
151
30.
178
May 29.
06 192
149
255
365
470
576
687
187
141
128
107
082
064
June 27.
527 069
630
738
442
347
252
155
043
022
102
130
153
176
June 30.
104 179
10
88
180
273
368
462
558
July
3
8
5
1
6
188
175
159
148
113
093
063
5.
070
098
123
141
166
173
185
200
176
139
104 DR J. HALM ON SPECTROSCOPIC OBSERVATIONS OF THE SUN.
Table IV. — continued.
Lat. Vel.
1902.
July 5
442
54 4
649
65 3
755
contrf.
102
083
058
056
029
Aug. 21.
498 100
Aug.
598
502
415
330
238
147
Aug
59"
25
102
177
23.
067
092
116
123
149
173
25.
171
154
175
166
Aug. 26.
356
249
112
07
224
36
371
•41
•65
•65
•73
■69
•78
•44
Aug. 27.
64 172
Aug. 28.
82 192
172
257
348
436
528
615
705
787
784
185
166
144
131
096
069
040
015
037
Aug. 29.
744 041
Lat.
Vel.
Sept
791
. 8.
039
742
013
640
063
535
089
429
114
304
140
197
165
89
180
Sept
186
9.
172
80
186
Lat. Vel.
Sept. 13.
29 181
110 186
Sept. 17.
122 190
14 201
94 185
Sept.
99
10
69
151
211
298
23.
180
195
189
177
171
152
Oct. 4.
367 150
455
117
545
108
624
071
708
033
Oct.
19.
21
170
45
189
133
176
219
164
307
150
394
117
483
102
Oct.
621
704
783
790
30.
057
029
013
048
Nov.
800
720
Nov.
7
1.
041
053
4.
084
105
132
154
164
161
Nov. 10.
777 029
1903.
Feb. 25.
240 165
388 130
Feb. 27.
354 152
172 187
94 194
Mar.
145
2 3
89
205
318
528
631
15.
183
202
199
200
168
119
087
Apr. 17.
72 185
183 178
293 165
Apr.
163
283
Apr.
99
208
316
419
530
640
743
838
817
715
19.
179
160
24.
213
201
170
134
103
074
042
016
024
053
Lat.
597
486
May
39
71
178
285
387
490
597
709
813
748
643
537
431
325
May
466
363
259
150
39
68
173
279
384
489
596
699
May
648
753
863
831
729
615
510
405
28'8
184
77
28
131
Vel.
081
103
23.
184
186
180
165
129
117
077
053
026
048
075
104
135
146
26.
•25
•41
•69
•77
■91
•83
•93
•74
•54
•20
•99
•63
27.
089
056
019
014
033
079
102
135
161
168
184
191
193
May 28.
44
151
259
361
473
576
681
•93
•86
•66
•46
•26
•95
•53
Lat. Vel.
Ma 1
{ 29.
728
049
83 3
024
859
014
758
033
654
068
548
082
442
113
340
156
235
173
132
178
29
184
77
173
June 3.
55 1 82
41 184
135 167
June 12.
206 195
301 163
396 131
June 20.
333 144
429
525
621
715
812
792
699
126
107
0"77
033
009
055
052
Aug. 14.
414
1
319
1
230
1
135
1
44
1
51
1
141
1
234
1
323
1
Aug.
400
481
574
661
747
812
755
Aug.
692
605
46
41
69
88
98
90
96
81
67
17.
168
134
110
075
058
031
014
18.
039
065
Lat.
51 2
423
329
235
245
Vel.
092
113
147
156
168
Aug. 27.
19 181
Sept.
709
788
Sept.
167
300
384
444
537
14.
056
009
15.
159
136
128
128
100
Sept. 18.
735 073
640
549
46'0
376
284
193
100
02
Sept.
109
194
286
372
46 3
525
601
688
773
Sept.
669
Oct.
675
750
818
805
Oct.
660
787
836
798
092
113
117
157
173
172
193
196
19.
192
166
158
133
120
120
087
042
023
27.
064
11.
076
033
028
020
13.
078
032
016
046
Lat.
700
591
472
359
251
Oct.
181
64
55
174
Vel.
055
071
120
145
167
21.
190
194
183
184
Oct. 23.
112 194
32 185
Oct.
119
207
293
376
425
506
585
Oct.
672
563
443
334
213
98
Nov.
68
156
239
324
404
488
572
27.
179
172
161
142
116
097
077
31.
068
094
128
149
151
182
3.
182
189
166
157
138
111
104
Nov. 4.
31 199
120 185
205 182
271 160
Nov.
410
499
581
654
738
816
860
6.
157
128
101
069
057
033
016
( 105 )
VI. — Theorems relating to a Generalisation of the BesselFunction. By the Rev. F. H.
Jackson, H.M.S. " Irresistible." Communicated by Dr W. Peddie.
(MS. received February 17, 1904. Read March 21, 1904. Issued separately May 27, 1904.)
1.
In this paper, theorems which are extensions of the following, are discussed : —
J (x + y) = J (x) J (y)  2J 1 (x)J 1 (y) + 2J 2 (x)J 2 (y)  ad inf. . . . (a)
1 ={J {x)}*+2{J i (x)}* + 2{J s (x)}* + adinf (/3)
t i \ i i\»It a„\ 2m(m + n) T . . , 2 2 m(m  l)(m + n)(m + n  1) T , N  , >
\ X 2] x z I
S_ n (x) = (lfJ n {x)
t / \ / ^^n{^) n+t d n (sin a;) ,,.
We define J [n] (A, x) as
r=oo ^
£>[r~]\[n + r}\{2) r {2) n+r
In this expression
[n + r]\ is T p ([n + r + 1]) or II p ([> + r)
r , • B"  1
^" + " L ^ J iy[»+7TI])
The function IIp([n]) is defined in the previous paper (Trans. Roy. Soc. Edin., vol. xli.
part i.). If A be changed to iX in J [n] , the function will then be more strictly analogous
to J n .
In Weierstrassian form
— ~ = \xy x "u i f l + »*m V^ ] !■
r p ([a]) L J „=i I \ 1 [n]J J
P=l +r l + L+ log?
[3] T [S] >l
{2n}l(2),,r,([ n .+ l])
being in the case of n positive and integral
{2»}l[2][4][6] . . . [2n]
analogous to
2 . 4 . 6 . . . . 2n = 2".n\
This notation enables us to write shortly
l lV»'\"+2r
d[n] 2 J {2« + 2r}!{2r}!
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6).
106 THE REV. F. H. JACKSON ON
2.
If we invert the base element p, we see that [r]! is transformed intop"'' (r " 1)/2 [r]! and
that (2) r becomes transformed into p' r(r ~ 1),2 (2) r . These transformations hold whether r
be integral or not. Inverting the base p in the series J [n] ft), we obtain
„«*v (W (n+r ' AY +ar a)
P H[r]\[n + r]l(2) r (2) n+ XpJ ' ' ' ' {>
which we denote
y 2 i M () (2)
Lommel has shown that
'm + n + 2r'
<wua) = 2/i)"_ _i_ _2 (J) ... (3)
'•=" T(m + r+l)T(n + r+\)
The function J [n) was formed while seeking to extend the above theorem. The
extension was found to be
JwWImW JwWiwW Z. ol>([m + re + r+ l])r p2 ([m + r+ l^r^n + r + l])r p2 ([r+ l])l [2]^ V '
The relation between J and J was surmised from the following simple but similar
theorem.
4W  1 + rTT, + m +
[1]! [2]l
■lW  i + £p + ^ +
w«  >  S + pf +
which suggested that J [n] might be derived from J [n] by inverting the base p. As I
have given the proof of (4) as an example of the use of generalised Gammafunctions in
a paper communicated to the Eoyal Society, London, it will be sufficient to say here,
that the theorem may be proved by showing, that the coefficients of the powers of X on
both sides of the equation are identical, being cases of the extension of Vandermonde's
theorem (Proc. Lond. Math. Soc, series 2, vol. i. p. 63).
In the notation of Art. ( 1 ) we may write the theorem
JwWImW = ^{2m + 2?i + 2r}\{2m + 2r}l{2n + 2r}\{2r}\ Xm+ " +r ' ' ( 5 )
Consider the series
•i l n l (A)„ 1 A) + fflj m (x) [1] (A)+ +jp s  8  1 [Sj w (X)I w W+ (6)
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION. 107
By means of (5) we write this —
h • MM. >2 + ( l)r W X «r + \
I [2] [2] [2] [2] {2r}!{2r}!{2r}!{2r}! + J
[2] I [4]"[2] [2] [2] {2r + 2}l{2r}l{2r}l{2r2} i w
9 r.MMj t4r^ A2r +
[2r] I i4r}l{2r}I{2r>I{0}J
We see by inspection that the coefficient of A 2 vanishes : the coefficient of A 2 '' is the
expression
n y W h_W J2d_ + «J23 [2r][2r2]
V ; {2r}!{2?}!{2r}!{2r}! I [2] [2r + 2] 1 [4] [2r + 2][2r + 4] ' ' '
,/ 1)y M,M [2r][2r2] [4] [2] \ (
+ { )P [2r][2r + 2][2r + 4] [4r] j ' K>
The series within the large brackets is easily summed term by term. The sum of the
first two terms is —p%— i, which is a factor of the third term. The sum of
L [2r + 2]
— ttz d. and the third term is
^[2r + 2]
[2r4] [2r2]
jP [2? + 2][2r + 4]
Continuing in this way, we obtain that the sum of the first r terms is
/ _ nr  1? ^ 1 , [2r2][2r4]l2r6] [4] [2]
V ' l [2r + 2][2r + 4][2r + 6] [4r2]
which is equal to the last term, but is of opposite sign. The coefficient of X 2 '' is zero,
and only the constant unity is left. Therefore
1 = J[0](X)l ra (A) + tjj [1] (A)l w (A)+ +P" 1 JJ [ #) 1 #)+ (9)
which, if p = 1, reduces to
1 = {J p + 2{J i p + 2{J 2 P+ (10)
4.
Consider now the series
JraMfmW[^J w (A.)I ra (A.)+ +P r ' r ^MM*)  (H)
Referring to expression (8), we see that the coefficient of A 2 " is
(If, , , W . [4] _[2jj_ [8] [2r][2r2] ,
V ' {2r}!{2r}!{2r}!{2r}!} X + [2] [2r + 2] + ^[4] [2r + 2][2r + 4] J V '
108 THE REV. F. H. JACKSON ON
The series within the large brackets, although simple in form, offers considerable
difficulty in summation. The sum is
2(p» + l)(p*+l) Cp**+l).(p» + l)(p*+l) . . . (^+l)l^iMl. . (13)
The sum in general for all values of r is
[2f [4r] r, 2 ([2r + 2]) ' ' ' { >
The reason that the simple series (p= l)
4 _2r_ 8. 2r(2r2)
2 2/ + 2 4 (2r + 2)(2r + 4)
is easily summed as
2 "2*
while the general series offers difficulty, is that the functions T p are present, both to
the basep 4 and the base_p 2 in the series (12) and in expression (14).
Heine has shown in his " Kugelfunctionen" that
^,b,c,p,x\l + x+ x +
„_.<!  bxp n )(l  p>»)
= n
6 ,bx,p,
e b
(15)
n=0 (l»p»)(]cp»)
■Consider now the series
[T][r+I] + ^[2][r + lJ[r + 2] + * [.] [r + 1] [r + 2] . . . [r + s] + ' " ' ' (lb)
Since
[1] i?l
W =^zi =^ + l
[2] J5 2 l ^
we write series (L6) as the sum of two Heine's series
b i + ^+i _ 1 & 2  + jfH _ j +^ r+I _ 1)(pr+2 _ l} + ■■ ■ ]
V +1  1 ' f +2  1 (y +2  1) . . . ('P r+s+1 i)i • ( 7)
These series we transform by means of (15).
First, for the series S x we put
a=p
b=p~ r
c=p r+1
x= jf
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFTJNCTION. 109
and obtain, after obvious reductions,
_ 2(1 + P )(l +p*) (1 +p*>) j P *  1 p (p*l)(p**l) P *
(1 y +1 )(l ~P r+i ) ■ • ■ ■ (1 ~P 2r ) t PI 2 (i?' 2 l)(^ 2 l) 2
(p»l)(p**l)(p*r*l) pj ) (lg)
(j> 2 l)(l> 8 l)(p 4 l) 2 ■•■■[
In the same way, if we put
we obtain, after reductions
a=p
b = p l ~ r
c=p r+2
X= — p r+1
s,i + ^:^ +
v r  1 i ,
— 7S P '
p r+i  r
+ff 2 )(i +p 3 ) (i +p r ) / 1 . p 2 "' 2  1 J: , 3 , (p 2r ~ 2  i)(^ 2r  4 „ 8 _ I (i9)
(i^ +2 )(iy +3 ) (ip 2r )t (pi)(p 2 +ir (pi)(p 2 i)(p 2 +i)(p 3 +ir ""<
T 1
So that S, +p p ~ S 9 may be written
p' + 1 "
{ i+p)(i+p*) (l+y 1 ) ( „ _.P 2r i , (p 2 i)(/ r  2 iU _
(l ^ +1 )(i y+ 2 ) .... (i i? 21 ) I p  1 7 (p 2  1)(/> 2  1)
p*l (p 2 D(p 2r  2 l )„4_ I . (20)
p+1^ (P 2 I)(i> 2 +1) l J
Adding the terms with like numerators together, we obtain
(1+P)q+P 2 ) (1+P' 1 ) I 2 _ 9b2 /^l . o, yB (P 2 "  l )(f r ' 2 7 !) _ 1 (21)
(l p r+1 )(l ^ r + 2 ) (l _p*) I " ^ r  I 2 (p 2  l)(p 4  1) ' )
The series within the large brackets is the simplest type of series, and its sum is well
known to be
2(1 jj»)(1 p+) (1 jF) .... (22)
We have therefore
i + EIJzL. + J£ Mfr 1 ] +
[!]['■+!] f 2 ] [' + !]['■ + 2] " ' "
2(1+^(1+^) . , , (1+J/1) • (1 ^1^) . , . {lp*)
(l;/+ 1 Kli/ + ') (l~P 2r )
Changing the base p to p 2 we obtain the series whose sum was sought
1 + [*] JM. 2 [8] [2r][2r2]
[2][2r + 2] ^[4][2r+2][2r + 4]
= 2(l+jJ 2 )(l+j? 4 ) ■ . (1+jr" 2 ) • (lp 4 )(lp 8 ) ■ • . (l~P 4r )
(1 y +2 )(l p r+i ) (1 p ir )
(23)
(24)
The coefficient of \ 2r is obtained by multiplying this sum by
W
{2r}! {2r}! {2r}\ {2/}!
110 THE KEV. F. H. JACKSON ON
which gives us
nr 2(l+^)(l+y)(l+y«) . . . (1+F 2 ) • (l+j>*)(l+^) (1+^Q (25 ,
[ ' {2r}!{2r}! ' ' V ;
as the coefficient of A 2r . If r be not integral, the infinite products in Heine's trans
formation do not reduce to finite products but to expressions in terms of the F p
functions, ultimately giving the sum of the series in the form (14).
Having obtained the coefficient of A 2r , we have established, subject to convergence,
the theorem
JraWIraW  [2]j w (^)I [ u(^)+^ 2 [jj [2 ](^!m(^)+
_ 2(1 +p*)\? ( iy . 2(l+^) . . . (l+y 2 )(l+^) . . . (l+f)X» ,
" " {2}!{2}! + ( } {2r}!{2r}! " (26)
which is the extension of
J (2X) = {J (X)P2{J 1 (X)P + 2{J 2 (X)P (27)
This is a particular case of the addition theorem for J .
J (A + A 1 ) = J (A)J (A 1 )2J 1 (A)J 1 (A 1 )+. . . .
5.
Defining * Sin p (A) and Cos p (A) as
Sin„ W = X !+  r .,..
we obtain
Sin, (A) Cost (A,) + Cos, (A) Sum (A,) = (A + A,)  (^ + W + V 2 ) (* + V 4 ) + ....
p p [dy.
Sin, (A) Cos x (A) + Cos,,(A)Si ni (A) = 2A _ ^(1 +^)(1 +i^ 3 +
p p [3J!
This suggests that the extension of the addition theorem of J (^ + \) will be on
similar lines.
Consider now the series
JraWI ra (A 1 )jj w (A) [1] (A l ) + (l)'y"' 1 'Hj [ i](A)l w (A 1 )+ (28)
I'ioc. Ed,in. Math. Soc, vol. xxii., 1904.
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION. Ill
and the product
J[»](^)Im(\)
 J x n ^ . , ' *l +2r _ U V  + K +2r P 2r(n+ r) _ I m s
I {2ra}! {2ra + 2}!{2}! '" {2n + 2r}\ {2r}\ ' ' ' I I {2n}\ {2n + 2r}\ {2r}\ ) K '
From series (28) we are to form a new series, of which the successive terms will be
homogeneous in AX X and of degrees 0, 2, 4, 6, 2r, . . . , . respectively.
The first term of (28) gives rise to the constant, unity.
The terms of the second degree arising from J [0] J [0] are
A 2 A % p 2
"{2]TT2}! ^ "{2M2}!
The term of the second degree arising from J[i]Jf[i] is
[2] {2}! {2}!
There are no other terms of the second degree ; the sum of these terms is
_ (A + y 2 )(A + A 1 )
{2}! {2}!
Terms of the fourth degree arise only from the first, second, and third terms of (28),
heing respectively
a 4 A^y \y
{4}!{4}! + {2}! {2}! {2}! {2}! {4}! {4}!
[£U *%__ + A W  I . . (30)
L2]}{4}!{2}!{2}! {4}!{2}!{2}! f * >
y [4]{4}!{4}!
Remembering that
{4}! = [41[2] and {2}! =[2]
we write the sum of (30)
A4 . M l^)\3\ . W_W>2\ 2„2 , [S]\2A 2 W 2 4. [ilM A A 3 4. „tS\ i \
[4] [2] [4] [2] 1 '[2] [2] 1T [2][2] ^'[4] ^'[2] [2]
Replacing
W by (jps + l) and ® by p* + l
the expression within the large brackets reduces to
(A + A 1 )(A +p%)(\ +p%)(X+ P %)
The term of the sixth degree in X, \ I have verified as
~ L6J L4] [2] 1 [6] [4] [2 ] { (A + Xl)(A + X ' p2){X + V * } * (X + Al?,2)(A + Xlpi){X + Xip6 } ' (31)
112 THE REV. F. H. JACKSON ON
The term of degree 2r is the following expression —
I 2r)!{2r}! 2r  2}!{2r  2}!{2}!{2}! I2r}!{2r}! 1
+ Wi
A 2 '^,
A''y^
[2]( {2r}!{2r2}!{2}! {2r2}!{2r4}!{4}!{2}!
A 1 2r 'Ap 2 '" 1)
+_p :
2 [s]( A*y
., .+
A' v y/^
[4] I {2r}!{2r4}!{4}! {2r 2}!{2r 6}!{6}!{2}!
. . . +
{2}!{2r2}!{2r}
{4}!{2r4}!{2r}! J
M! 1 • (32 >
[4r]f __Ay_)
^ f2rl U2r}!{2r}!j
[2r] I {2r}!{2?}!
We have shown in Art. (4) that in case A = X 1 this expression is
(A + A)(A + Aj) 2 )(A + Aj; 4 ) . . . (A + Ajr' 2 )(A + Ap 2 )(A + A^) (A + Ajr r )
{2r}!{2r}!
(33)
It has been directly verified that for particular values of r (l, 2, 3) the forms, in case
X be not equal to \, are
_ (A + A 1 )(A + A 1 p 2 )
{2}!{2}!
(A + A t )(A + A lP 2 )(A + A^)(A + y*)
{4}!{4}!
_ (A + A, )(A + A ^XA + \ lP *) ■ (A + A 1 j/)(A + A^)(A + A 1 p«)
. {6}!{6}!
respectively. This indirectly establishes the form of the coefficient of degree 2r in
A and \. A direct proof of the algebraic identity would, however, be preferable.
Writing now
£f >F j^^WAi)  i + Z.V^ {2«}!{2w}!
(34)
If p = 1 , we obtain the addition theorem of J
JofA + Aj) = J (A)J (A 1 )2J 1 (A)J 1 (A 1 )+ , . . (35)
6.
The analogue of Lommel's theorem
j.w = (irj^) 2 ^ ) w 1 w+ }
I have shown by two distinct methods * that
t (py~« l){ 2)y~ a+1  1) . . . (py P +'i l). (pyPl)(pyP+ l  1) (jjy£+«i l)
K ii (^y«/»l)(pv*0+ 1 l) . . . (#■ M«ii) (pvl)( j ijr+ 1 l) . . . (pv+*il)
p Ii][y] ' [iJ^]M[y+i]
(36)
* Proc. Lorad AfaiA.. Soc, series 2, vol. i. pp. 71, 72, 1903, and Amer. Jour. Math., vol. xxvi., 1904.
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION. 113
In terms of the function T p of this paper, this theorem is
J_ iy[ya/3] )iy[ y ]) = [a]j£j
^ r y ([ 7 a])r4 7 :/i]) + ^[i][y] " " " " ' ' (6i}
Change the base p to p 2 and put
a = — m
f3 = mn
y = r—m+1
we obtain
IVflr + m + n + l])iy,([r  m + 1]) = 1 + [2m] [2m + 2» ] y2 _ 4ffl _ 2re (
I>(> + l])r^([ r + »+!]) t 2 ] [2r  2m + 2f
[2m] [2m  2s + 2] [2m + 2m] [2m + 2n  2s + 2] ^_ 2t(2OT+ „, ,„„,
[2] [4] . . . [2s][2j2m + 2] . . . [2r 2m + 2s] F y '
Now consider
. .. p» [2m ][2m + 2»] T ... f [2m][2m  2][2w + 2»][2m + 2n  2] T ,., ,
Wn]W "^ShS L ~ JL  2 j A J J Pm +nvW + pSm+in L j 2 ][ 4  J «W.«W + • ■ ■ ■ (39>
The coefficient of X' l+2r is the infinite series
Iz 1 )^'" [~i+ p2 [MI2^±M + 1 uo)
{2n + 2m + 2r}l{2r2m}\i p im+2n [2] [2r  2m + 2] J ' " * y
which by (38) reduces to
(1)''" I>([m + » + r + 1 ])r^([r  m + 1])
{2n + 2m + 2r}l{2r2m}\ I>([r + l])I>([r + n + 1])
Now remembering
{2s}! = [2]T i)2 ([,+ l]) = (2) s r,([.+ l])
the expression (40) reduces to
r7([r + l])r,([ W + r+l])(2),.(2)„ + ,.
(41).
■n
(_iy» P . . (42).
which is (  i)»p2m ( m+n) x coet }icient of \ n+2r in the series J [n] .
This establishes
(  iyy m,m+ "»j M (A) = j [2m+n] (A)  _£. C^[2^±2»] J[2m+B _ u(X) + — ad inl . (43) .
an extension of
j„ = ( ir < j»h.  2 ^ ± zL ) J 2M+n  1 + } • • (44).
Lommel defined J n for negative integral values of n, so as to make this theorem
always hold : for examp]e, suppose n a negative integer, and put it equal to  m, then
we have by this theorem
j [ _ n] = (irj [n] (45>
extending
J_„ = (1) W J„
also
![_„ = (i)*l M
as may be shown by inverting 'the base p in expression (43).
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 18
114
THE REV. F. H. JACKSON ON
7.
If we now define
Sm p (\,x) as A*  _. + — _
(46)
then
Sm„(A,^) 1 _ A 2 a: [2]
Operating on this with g [n] which is
I d
\2n+2r„.[2n+2r]
K ' [2»+2r + l]l
' 1 d(x^) \ d{aF) ) ] )
(47)
the first n terms of the series are destroyed, while the term involving xP n+2r] is reduced
to
T2r+2l o
Taking
, _ 1 x n+ r A 2«+2r 1 [2w + 2r][2« + 2r2] . ■ .
^ ; [2]" [2n+2r + l]l
(48)
^n+j+2r
a ;[«+J+2»]
L« + r + J]![r]!(2) r (2), 1+9 . +i
[n + r + fll = [» + , + *] [n + rJ] [f]r,([]
(2) n+r+} = d*^» + l) ( 2 <+'»+ 1) .... (pf + 1) • (2),
and
therefore
So we obtain
J [n+i] {\,x)
(2) i r,([i + ]) = [2]»r^([l+i])
[n + r + fi\(2) n+r+i = [2rc+2r+l]
,2( 1 )"r,
. . [3] x [2]iU P 4i])
[2]*T^[f])^ x '[2 W + 2r + l] . . . [5] [3] • [2] [4] . . . [2r]
and by a change of the variable x
A J [TO+H (A,a )  ( 1)  [2]I>([f]) A 1 j(F" J '
A w denoting the operator
(49)
(50)
(51)
(52)
For further properties of Sin p , Cos^, and their connection with symbolical solutions
of certain differential equations, reference may be made to a paper on " Basic Sines and
Cosines" (Proc. Edin. Math. Soc, 1904).
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION. 115
Continuation op Paper —
" Theorems relating to a Generalisation of the BesselFunction."
(MS. received April 19, 1904.)
8.
The theorem
jmm»)$m%m+ +(i)y (s  1, Bj]jc S] («)i M w
1 _ (a + b)(a + bp 2 ) (a + b)(a + bp 2 )(a + bp 2 )(a + 5p 4 ) _ , ,
[2] 2 + [2] 2 [4p ~  ' * {a)
discussed in the first part of this paper may be obtained very naturally from the
properties of a certain function analogous to the exponential function. Elsewhere," 3 " by
means of the function E p I have obtained
Wh(j)~WMh(j) + +(~mP*Jd*)h(j)
~W~ Pit*? ' w
We naturally expect to find some general form to which both (a) and (/3) will belong,
as particular cases. The following is the general theorem which will be obtained from
the function E p , just as the addition theorem for Bessel coefficients is obtained by
means of the exponential function. Exp. ( ■xi t —  J
^(a.&JJ^a)!^^ (y)
[2vJ >J
T" /„ h\  l ( a + t>)(a + bf) (a + &)(« + ty 2 )(« + 6p*)(a + &p Sr+2 )
[2] 2 T " _ [2]W
In case v = 0we have the quasiaddition theorem (/3). If, however, v= 1 we have the
quasi  addition theorem (a). The corresponding theorems for the function J* (a , b)
will be briefly noticed.
y , h) _ (a + b)(a + bp*) .... (a + bp**) ( , _ {a + bp*)(a+bp>)
°^ a ' 0) [2] [4] . . . [2ra] 1 [2» + 2][2] "
(a + fy? 2 ")(a + bp 2n+i )(a + fb)(a + ;r" +2 6) _ I (S)
+ ~ [2n+2][2n + 4][2][4] ~ ' ' ' J * j
The expression for J* (a , 6) will be given also in the case when n is not a positive
integer.
* Proc. Lond. Math. Soc, shortly to be published.
116 THE REV. F. H. JACKSON ON
9.
In this article certain results will be obtained which will be required in subsequent
work. We define the function E p (a) as
E *)* 1+ [T]! + r2]! +
If we invert the base p
E i« 1+ [rp + *[J + ■ • • • + ""'% + ■ ■ ■ ■
without difficulty we have
E»EL(*) = l + (ffi + (« + »>fr + »> + (e)
Changing p to p 2
M.»h(»)i+^+^ X '^ + ....
p 2 — 1 /o 2  1 p 2 — 1
_ [2](q + &) [2]*(a+&)(a + 6p*)
[2] + [2] [4] "+••■•
M[2]JM[2]J = 1 + [2T + [2] [4] + (r?)
In part (l) we have established
J t _ n] («)= (l)"J tB] (a) (6)
Inverting the base p we obtain also from this
l^/a) = (  in M (a) ( K )
10.
A consideration of the product of the two absolutely convergent series
w ( at\ , at u 2 t 2 . a"t n
M[2]; = 1+ [2] + [2j[4] + ••■• + {m> +
E A[2]; 1 [2T + [2][4] •••• (  1) TM! + ••••
shows us that
E *(^) E *($j) = 2J M («r+:(i) B J M («r' 1
= VJ w (a)f (A)
In precisely the same manner, if we consider the product of
*W + [2] + ^L2][4] + • •• • + ^ {2w }. + '■■■
A [2j ; [2] +7 [2] [4] ( ^ {2rc}! +
* Pvjc. J'.clin. Math. Soc, vol. xxii.
THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION. 117
we obtain
El (m) El ( J ^') = Zp'^Mb^r+Zi itf^mw 1 )*"
v 2 \*y p'\ L^J ' »=o »=i
= Jj> ntn "%m(W 1 )t n (/*>
We have now, on taking the product of (X) and (m),
+» +« / at \ / at~ i \ / bt \ / '» 2 "6^ _1 \
2Jw<«)<" x Z^ < """ , Im(^' 1 )^ = M[2])M"T2j) E ^([2]) E .( ~ IXT )
The product of the four basicexponential functions on the right of this expression is
the product of two convergent series
/ , , (a + b)t, (a + b)(a + p*b)f 2 , I „ J n (a+P 2v b)t^ , (a + p»b){a + p»+* b) t^ ) , .
i 1+ [2] + — T2]W rM 1 "— [2T~ + p^KT" ") ()
This result follows from result (>?) of article (9).
If now we equate coefficients of the various powers of t in (B) with the corresponding
coefficients in
+ 00 +00
remembering that
J[«] = (  1 )"J[«]
3 M = (!)"![»]
we obtain from the terms which are independent of t
JraHIra^jp" 1 )^ (p)<
= , _ (a + b)(a +p"b)
~w~
which by an obvious reduction becomes
Jto](«)3 [ o ] (^" 1 ) y^Jral")!^ 1 ) +..••+( W^'&JMMW 1 )  (cr>
= J* (a,b)
Equating the coefficients of t n we obtain
m=+x
Z J[».](«)[»».](^ , '" , )?' , "" ,, " , """ , ""=J;(fl.6) ■ ■ • (t>
m=— oo
the expression for JT being that given in article (8) expression (5).
11.
When n is not a positive integer the expression
(a + b)(a+2) 2 b) (a +p^~ 2 b) in J* (a , 6)
must be replaced by
J_j (a + &)(a+jr&)(a+jj 4 fr) .... (a+jfozft) „
«=»(a+y6)(a+_p 2 »+ 2 6) .... ( a +_p2n+at2j) a
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 19
118 THEOREMS RELATING TO A GENERALISATION OF THE BESSELFUNCTION.
If. however, p > 1 ,
T ( a + bp^ia + bp 2 **) . . . (a + Sp 2 " 8 ") ^
«ii (a +p i b)(a + p~ i b) . . . {a+p 2 «b)
is the effective representative of the product (a + b)(a +p*b) .... to w factors. This
corresponds to the change of n\ in the Bessel coefficients into T(n+1) in the case of
BesselFunctions. The series expansions of the products given above may be found in
Proc. L.M.S., series 2, vol. i. pp. 6388. The theorem analogous to Neumann's
theorem
J ( cl * + V + 2ab cos 8) = J (a)J (b) + 2^(l) s J s (a) J. &) cos s6 . . (I)
I have investigated in a paper (Proc. L.M.S.). The function E p being used in a
manner similar to the use of the exponential (pp. 25, 26, 27, Gray and Matthew's
Treatise on Bessel Functions), gives us a rather complicated extension of (£).
( 119 )
VII. — On Some Points in the Early Development of Motor Nerve Trunks and
Myotomes in Lepidosiren paradoxa (Fitz.). By J. Graham Kerr, Kegius
Professor of Zoology in the University of Glasgow. (With Six Plates.)
(Read January 18 ; MS. received February 9, 1904. Issued separately July 1, 1904.)
CONTENTS.
Introductory 119
Development op the Motor Nerve Trunks . 119
Development op the Myomeres . . . 122
PAGE
General Remarks . . . . .125
Explanation of Plates . . . . 127
Introductory.
My main purpose in the following short paper is to publish figures illustrating some
of the more important facts of the early development of myotomes and motor nerves
in Lepidosiren. The bearing of some of the observations of nerve development upon
current theories renders it particularly desirable that they should be illustrated by
untouched photographs of the sections. A few photographs illustrating the more
important stages * in the development of the motor nerve trunks are given on Plate I.
For the preparation of the photographs here published, as well as several others, I am
indebted to the skill of my friend Dr T. H. Bryce, and of Mr Fingland, our University
photographer.
Development of the Motor Trunks of the Spinal Nerves.
In describing the observed phenomena it will be convenient to begin with a late
stage in development and work backwards to the earlier stages, and so pass from the
better known and more familiar to the less known and less familiar.
Stage 34. — Fig. 1 illustrates a considerable length of motor nerve from stage 34.t
Here the nerve (n.t.) consists of a cylindrical mass of nerve fibres about 13/u. in
diameter. On the surface of this the nuclei of the protoplasmic sheath are conspicuous.
The greater part of the sheath itself is so thin as to be invisible even under the 3 mm.
immersion objective, except in the neighbourhood of each nucleus, where it swells out
to form a thick mass containing the nucleus.
Stage 31. — At this stage the nerve rudiment on superficial examination presents
the appearance simply of a chain of nuclei placed end to end in a strand of protoplasm.
* By stage n I mean the stage represented by fig. n in my paper on the external features during development,
Phil. Trans. Roy. Soc. B., vol. cxcii. p. 299.
t Cf. Rapfaele's fig. in Anat. Anz., 1900, p. 340 (Per la genesi dei nervi da catene cellulari). Of. also Kolliker's
remarks on this, op. cit., p. 511.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 7). 20
120 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE
Appearances, in fact, support the Kettentheorie — suggesting a chain of " cells " placed end
to end. More careful examination shows the presence within the protoplasmic strand
of a cord, faintly fibrillated longitudinally, and differentiated from the simple proto
plasm by its affinity for eosin. The nerve runs downwards along the inner face of the
myotome, and it is difficult to make out with certainty its connections with the cells of
the myotome.
Stage 29. — At this stage (figs. 2 and 7) the conducting part of the nerve (n.t.) is in
a similar condition to that described for stage 30 — a distinctly fibrillated band —
usually about 2 '5 to 3 m in thickness close to its root — which stains deeply with eosin.
It slopes outwards and downwards from the ventrolateral angle of the spinal cord to
the inner surface of the myotome, along which it proceeds in a ventral direction. The
protoplasmic sheath (p.s.), however, is now far more conspicuous than in later stages.
It is a great, irregularly thickened mass of granular protoplasm, sharply marked off from
the true nerve by its being stained more greyish in colour in hematoxylin and eosin
preparations, in sharp contrast to the deep red of the nerve trunk.
Scattered through the protoplasm of the sheath are large nuclei rich in chromatin,
yolk granules, and here and there vocuoles.
Stage 27. — At this stage (figs. 3 and 8) the most conspicuous difference from stage
29 is that the nerve trunk (n.t.) is now naked for the greater part of its length. At its
outer end it spreads out into a number of strands arranged in a conical fashion. In the
case of the strands near the axis of the cone — i.e. in the case of the strands which
pursue a direct course towards the inner surface of the myotome — it may be clearly seen
under a high power of the microscope that each strand passes perfectly continuously and
by insensible gradations into the granular protoplasm, which forms a taillike process of
a myoepithelial cell of the myotome. In the case of many of the motor trunks at this
stage there is to be seen a mass of mesenchymatous protoplasm (p.s.) richly laden with
yolk, and containing numerous nuclei, concentrated in the neighbourhood of the nerve
towards its outer end. This is the rudiment of the mesenchymatous sheath which in
stage 29 we saw had spread out over the surface of the nerve. The nerve trunk
itself is about the same thickness as in stage 29, though I find considerable variation
in this respect.
Stage 25 (figs. 4 and 9). — A little behind the middle of the body at this stage the
myotome is seen to be just commencing its recession from the spinal cord, mesenchyme
(me.) cells * richly laden with yolk having begun to migrate in between the two
structures. The nerve rudiment (n.t.) in the section figured is about 7m thick. It is
distinctly fibrillated, and at its lower end expands into a cone as in stage 27 — the base
of the cone, however, here being less expanded. Traces of longitudinal fibres are already
visible on the ventrolateral surface of the spinal cord.
* Here as elsewhere I use the word " cell " merely as a substitute for the more cumbrous expression " nucleated
mass of protoplasm " without in the least implying that it is separate from its neighbours. As a matter of fact the
" cells " of the mesenchyme are merely the enlarged and nucleated nodes of an irregular continuous protoplasmic
spongework such as Sedgwick describes in Selachians.
EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 121
Stage 24. — Transverse sections about the middle of the body at this stage (figs. 5
and 6) show the myotome still in contact with the wall of the neural canal, mesenchyme
either not having intruded itself between the two at all, or only having done so to a
very slight extent as yet except in the anterior part of the body. In embryos which
have been excised while alive in normal salt solution, spread out in one plane like that
figured to illustrate this stage in my account of the external features of development,*
and fixed in this position, the myotomes are pulled slightly away from the neural tube.
It is seen in transverse sections of such embryos that the rudiments of the motor nerve
trunks already exist as soft thin bridges (n.t.) metamerically arranged and connecting
spinal cord rudiment and myotome as shown in figs. 5 and 6.
The nerve rudiments at this early stage are formed of granular protoplasm either
without yolk or containing only very minute granules, without obvious fibrillar
structure when stained by the ordinary methods. The rudiment is quite naked, the
richly yolked mesenchymatous sheath of later stages being conspicuous by its absence.
Nor are there any nuclei in contact with the nerve rudiment. There can happily be
no doubt about this in Lepidosiren, where the nerve rudiment in this early condition is
of relatively small size compared with the dimensions of a single nucleus !
The nerve rudiment — composed as it is of soft protoplasm — is at first extremely
fragile, gradually becoming tougher as development proceeds. Consequently in a
straightened out embryo of this period we find the more posterior nerve rudiments — i.e.
those in an earlier stage of development— show a greater and greater tendency to be torn
away from the myotome in prepared sections. A nerve rudiment which has become
torn away from the myotome is shown in fig. 10, which brings out a point more difficult
to observe in the uninjured condition, that the protoplasmic mass forming the nerve
rudiment spreads out over the inner face of the myotome. How far this expansion
extends, whether — as is probable — it really covers the whole face of the myotome, is a
point almost impossible to decide definitely by actual observation. Similarly I am
deterred by the unreliability of observations made on a spinal cord so laden with yolk
in its early stages from saying anything as to the connections of nerve rudiments with
neuroblasts or other cellular elements in the substance of the spinal cord.
At this stage the spinal cord is without any obvious mantle of fibres.
The motor nerve trunk has thus been traced back to a period in which it is repre
sented by a bridge of soft granular protoplasm connecting spinal cord and myotome at
a stage when these structures are in close apposition. As the myotome becomes pushed
outwards by the development of mesenchyme, it remains connected with the spinal cord
by the everlengthening strand of nerve. As the nerve develops it soon loses its simple
granular protoplasmic character and assumes a fibrillated appearance. Eichly yolked
masses of mesenchymatous protoplasm become aggregated round the nerve, which till
now has been quite naked. At first this protoplasm forms an irregular mass towards
the outer end of the nerve trunk, but it soon spreads along it and forms a definite sheath.
* Phil. Trans. B., vol. cxcii, pi. 10, fig. 24.
122 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE
This is at first very thick and conspicuous, but it gradually thins out, its yolk is con
sumed, and eventually the only parts remaining conspicuously visible are the nuclei
dotted alon£ the surface of the nerve trunk.
Into the subsequent history of the motor nerve — which is of minor morphological
interest — I do not propose at this time to enter in detail.
The protoplasmic sheath grows into the nerve trunk, dividing it up into separate
bundles of fibrils. The nerve trunk, as has been shown, spreads out at an early stage
in conical fashion over the inner face of the myotome. As the myotome grows in
surface this cone becomes broken up into distinct strands which become more and
more divergent.
As the adult condition is reached the part of the nerve trunk proximal to the point
of divergence — i.e. to the apex of the cone — increases relatively little in length. The
distal portion, on the other hand, with its individual branches, increases enormously in
length and the branches become more and more changed in direction as the muscles to
which they are attached become pushed about by differential growth.
Development of the Myomeres.
The general features in the development of a typical myomere as seen in transverse
sections are shown in textfigures A to H, # and in detail in PI. III. and IV. figs. 1114.
The protovertebra, at first (textfig. A) a solid diverticulum of the enteric rudiment,
develops a myocoelic cavity through the breaking down of its central cells about
stage 21 (textfig. B).
By stage 24 (textfig. C) the myotome is beginning to show signs of a flattening
from without inwards. The myocele is obliterated, and the cells of its mesial wall have
become flattened in form. At their outer ends they interdigitate with the inner ends of
the cells of the outer wall, so that the line separating the two walls in a transverse
section is a zigzag one.
In stage 27 (textfig. D, PL III. fig. 11) the inner wall cells have become more
regular in shape ; forming rectangular parallelepipeds flattened in an obliquely dorso
ventral direction, so that their larger faces slope inwards and downwards. Contractile
fibres (fig. 11, c.f.) have now appeared in the body of these cells, running longitudinally
and forming most frequently a layer close to each of the dorsal and ventral surfaces, the
two layers becoming frequently continuous with one another externally, and sometimes
internally. Very often, however, the arrangement of fibres at their first appearance
does not show this regular arrangement. The contractile fibres appear to be rounded in
section, and are easily distinguished by their highly refringent character, and by their
peculiarly deep stain with Heidenhain's hematoxylin. The cells of the myotome are at
this stage still laden with yolk, and this naturally is a difficulty in the way of observation.
The muscle fibrillse are striped almost from the beginning. One can often see in
* These have been printed as separate Plates — V. and VI.
EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 123
Heidenhain hematoxylin preparations that the young fibril appears to be made up of
discrete particles arranged end to end.* But between these the protoplasmic matrix
shows a more or less distinct fibrillar structure, the fibrils causing it having a longi
tudinal course and being continuous. These continuous fibrils appear to be an earlier
stage of the contractile fibrils. The outer wall of the myotome in transverse section
has the appearance of being composed of a single layer of large cubical epithelial cells.
In horizontal sections the cell boundaries are less easy to make out, but when visible
they show that the cells form a truly cubical epithelium. During mitosis these cells
become more or less spherical.
Stage 31. — At this stage (textfig. E, and figs. 12 and 12a) the outer portion of the
muscle cell has increased in size, so that the outer limit of the contractile fibres is
relatively much further removed from the outer end of the cell than it was before. In
this outer end of the cell the cytoplasm assumes a clear transparent appearance, and in
the preserved specimens large clear vacuoles are seen which possibly in the fresh condi
tion contained glycogen. The inner part of the cell is now almost filled with contractile
fibres, the protoplasm being reduced to the matrix between them. In this matrix yolk
granules are still abundant, and it is noteworthy that the muscle cells are now becoming
multinucleate, the original nucleus having divided repeatedly. The division is mitotic.
As the divisions only take place at relatively long intervals, a little patience is required
in hunting for the mitotic figures. The resting nuclei lie free in the protoplasm of
the myoblast. It is interesting, however, that during the period of mitotic activity the
nucleus becomes surrounded by a sharply delimited more or less spherical mass of
protoplasm, simulating the appearance of a cell within the myoblast. So striking is
this appearance (PI. IV. fig. 14) that it suggested at first sight that more or fewer
of the nuclei of the myoblast were really the nuclei of cells which had wandered into
its substance from the mesenchyme without, just as such cells wander in later between
the muscle fibres. On the whole, however, the balance of evidence is in favour of the
celllike structure round the nucleus being merely a temporary phenomenon due in
some way to the influence of the mitotic activity of the nucleus on the surrounding
cytoplasm — a phenomenon of the same nature as the rounding off into a spherical shape
during mitosis of cells which in the resting condition are of more irregular outline.
The appearance of the myotome of this stage, as shown in horizontal sections, is
indicated in PL IV. fig. 12a.
At about this period a striking change comes over the outer wall of the myotome.
Numerous mitotic figures are observed in it. Its cells subdivide rapidly, so that the
outer wall becomes several cells thick. The innermost of the cells so arising become
squeezed in between the rounded ends of the primary muscle cells. At this stage it
is often difficult to draw a line of demarcation between the outermost cells and those
of the cutis which is now beginning to appear between myotome and skin. I am not,
however, prepared to assert definitely that they actually give rise to cutis.
* Cf. Godlewski, Arch. Mikr. Anat, Bd. lx., 1902.
124 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE
The outer wall cells now become converted into elongated irregularly cylindrical
cells which stretch continuously from one muscle septum to the next. In the proto
plasm of these there begin to appear about stage 31+ (PI. IV. figs. 13 and 13a, and
textfig. F) contractile fibrils of a similar nature to those which have been long present
in the inner wall cells. About stage 31 mesenchyme cells wander in between the
myotomes. These give rise to the substance of the septum. Some also wander in
between the muscle cells of the outer wall.
By stage 34 the layer of muscle cylinders arising from the outer wall much exceeds
the inner wall in thickness. It is distinguished from the latter at the first glance, its
muscle cells being slender cylinders instead of flattened parallelepipeds.
The general appearance in transverse section of a myotome of this stage will be
gathered from textfig. G. It will be seen that the lateral branch of the vagus nerve
with its surrounding mesenchyme has formed an immovable obstacle, so that the
myotome as it increases in thickness, and as it is pushed outwards by the increase of
mesenchyme on its inner side, becomes gradually divided into two portions, a dorsal
and a ventral, the two remaining for a time connected by a thin isthmus but being
eventually completely separated. Along the mesial face of the myotome are seen the
muscle cells of the inner wall, now reduced greatly in size in proportion to the whole
thickness of the myotome.
The whole thickness of the myotome outside this consists of the derivatives of the
outer wall. In its extreme outer portion it consists of cylindrical cells still in the
condition described for stage 31, in which contractile fibres are just beginning to
appear. From this in a mesial direction the muscle cells are seen to become of greater
and greater diameter, and their contained contractile substance increases, especially in
size, showing that the muscle cells are older as they are farther removed from the outer
surface.
This, together with the fact that mitotic figures are numerous in the external
layer, shows that this latter is the region in which growth in thickness of the myotome
takes place.
The myoepithelial cells of the inner wall remain distinct up to about stage 35,
though constantly becoming more and more insignificant as compared with the great
mass of the myotome composed of muscle cylinders derived from the outer wall.
About the stage mentioned, however, the myoepithelial cells begin to break down,
portions becoming segmented off from their outer ends. These resemble the muscle
cells of the outer layer in character, being long cylinders. This process goes on, and
soon the once conspicuous myoepithelial cells have become entirely resolved into these
cylindrical elements, and the myotome is composed of apparently similar elements
throughout its thickness (textfig. H).
EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 125
General Eemarks.
1. Motor Nerve Trunks.
As is well known, there are three main views regarding the development of motor
nerves in the vertebrata, which may be shortly stated as follows : —
(1) Each nerve fibre develops as an independent outgrowth from a ganglion cell
which gradually grows out towards, and finally and secondarily becomes united to, its
special muscle. The sheath of protoplasm surrounding the nerve is an accessory
structure of independent origin developed from mesenchyme. This view is associated
especially with the name of His, and is the view favoured by the majority of
embryologists.
(2) The nerve trunk is multicellular in origin, consisting at first of a chain of cells,
in the substance of which the nerve fibres are developed later, as fine fibres passing
continuously from one cell body to another. The elements forming the original chain
are most frequently looked on as ectodermal elements which have wandered out from
the spinal cord rudiments. The protoplasmic sheath is derived from parts of the
original cell chains which retain their protoplasmic character (Balfour, Goette,
Beard, Dohrn, v. Wijhe, and others).
(3) The nerve trunk is not a secondarily formed bridge between spinal cord and
motor end organ. It has existed from the first, and in subsequent development it
merely undergoes elaboration from its at first simple protoplasmic beginning (v. Baer,
Hensen, Sedgwick, Furbringer and others).
It is clear that the facts of development in Lepiclosiren, at least in the motor nerves
of that animal, give strong support to the lastmentioned view as regards the nerve
trunk itself, and to the second view as regards the protoplasmic sheath. It has been
shown that by the examination of earlier and earlier stages the motor trunk can be
traced back, without, I think, any possibility of error, to a simple protoplasmic bridge
which already connects the substance of the medullary tube with that of the myotome
at a stage when they are still in contact.
As regards the origin of the protoplasmic sheath the evidence of Lepidosiren is
equally emphatic. In its early stages the motor trunk is perfectly naked. About
stage 27 masses of mesenchymatous protoplasm kden with yolk become applied to the
nerve trunk, at first over only a small portion of its length, and these masses of
protoplasm gradually spread over the whole trunk, remaining, however, for some time
clearly distinguishable from the nerve trunk by their difference in staining reaction.
As development goes on the yolk becomes used up, the protoplasm with its nuclei
extends into the substance of the nerve trunk — doubtless to keep up the proper
proportion between the bulk of the nerve trunk and its nutritive surface in contact with
the sheath protoplasm. The protoplasm itself becomes less and less conspicuous, and
eventually is only to be detected in the immediate vicinity of the nuclei.
126 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE
Lcpidosiren offers no evidence, so far, as to the ultimate origin of the nerve fibrils.
They appear gradually in an at first simple protoplasmic matrix. They seem to form,
as it were, a kind of picture in morphological symbols of already existent physiological
activities. It would seem that the continued passage backwards and forwards between
centre and end organ of a stream of nerve impulses gradually finds expression in the
marking out of the original simplelooking granular protoplasm into definite fibrillar
impulse tracks,* the undifferentiated protoplasm remaining as the perifibrillar
substance. Such a view of the nature of the fibrils is supported by Bethe's remarkable
observation t that in the chick embryo about the end of the sixth day a nerve trunk
may lose its fibrillar character in the immediate neighbourhood of a mitotic figure, to
reassume it on the completion of mitosis.
It is pretty clear that the great function of the sheath is to serve as a nutritive
organ for the nerve trunk. We see how its protoplasm is at first laden with yolk which
gradually becomes used up as the nerve trunk develops within it.
That the main function of nuclei, apart from reproduction, is to control cytoplasmic
metabolism is well recognised. The nuclei of the sheath are able to exercise this
control over the active metabolism of the developing nerve trunk which is without nuclei
of its own. Connected with this relation of the sheath nuclei to the metabolism of
the nerve trunk is no doubt the active multiplication of these nuclei observed in
early stages of nerve regeneration.! In such regeneration it may well be that the
protoplasmic matrix of the nerve simply repeats the process of its original develop
ment, increasing in size and then developing nerve fibrils within itself. If these
fibrils represent merely the differentiated paths of nerve impulses passing through
the substance of the protoplasm, it would of course happen naturally that the
regenerated fibres would be formed in continuity with those of the undegenerated
stumps. On this view the process which takes place when the peripheral
part of a cut nerve degenerates and then regenerates is somewhat as follows : — The
fibrils, no longer subject to the stimulus of passing nerve impulses, revert to their proto
plasmic condition. The protoplasmic sheath becomes highly active. § It increases in
thickness : its nuclei divide actively. Its protoplasm digests the remains of the
medullary sheath.  It thus comes to contain stored up food material as in its original
embryonic heavily yolked condition. The protoplasmic matrix representing the
degenerated axis cylinder lies imbedded within the sheath.ir Controlled and supplied
with nourishment by the activities of the surrounding sheath the protoplasm behaves
just as it does in ontogenetic development: (l) it grows — probably slowly — and
* Were this the case, it might well be that the formation of fibrils might tend as 1 a rule to spread from the end of
the nerve from which came the most active and frequent nerve impulses.
t Bkthe, Allgemeine Anatomie unci Physiologie des Nervensystems, p. 244.
X Bungner, Ziegler's Beitrage z. Path. Anat., Bd. x., 1891.
§ Wieting, op. cit., Bd. xxiii., 1898.
 This view of nerve regeneration, which my ontogenetic work inclines me towards, appears to agree most closely
with that enunciated by Neumann {Arch. Path. Anat. u. Phys., Bd. clviii. p. 466).
IT This protoplasmic strand within the protoplasmic sheath could only be demonstrated with extreme difficulty.
EAELY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES 127
so gaps are bridged over ; and (2) as soon as it has become continuous, nerve impulses
beginning to play backwards and forwards in its substance cause again a differentiation
into fibrils. As part of the impulse tracks persist as the stumps of the fibrils, the
regenerate parts of the fibrils will naturally develop in exact continuity with these.
I have no intention of entering into a general discussion either as to the data in
connection with the development of nerves in other Vertebrates, or as to the general
conclusions which have been based on them. My purpose now is merely to emphasize
the observations of the phenomena as they occur in Lepidosiren. I may, however, be
allowed to point out that such observations as those of Bethe upon the Chick, * although
apparently supporting the cell chain view, are in no way irreconcilable with the
observations here chronicled. The soft protoplasmic bridges which form the first
distinguishable rudiment of the motor nerve are difficult to observe in Lepidosiren,
whose histological features are upon a relatively large and coarse scale ; and how much
more delicate and difficult to observe are the corresponding structures likely to be in
the chick ! It may well be that further research will demonstrate the existence of a
delicate protoplasmic bridge about which are clustered the " nerve cells " (in Apathy's
sense) observed by upholders of the cell chain theory.
DESCRIPTION OF PLATES.
General List of Abbreviations.
c./. Contractile fibrils. nu. Nucleus.
d.r. Dorsal root. n.c. Neural canal.
g. Glycogencontaining outer portion of myoblast. n.s. Nuclei of protoplasmic sheath.
i.w. Inner wall of myotome. n.t. Nerve trunk.
l.vag. Lateral branch of vagus. o.w. Outer wall of myotome.
me. Mesenchyme. p.s. Protoplasmic sheath of nerve trunk.
m.n.r. Motor nerve root. s.c. Spinal cord.
my. Myotome. v.r. Ventral root.
n. Notochord. y. Yolk granules.
Plate I.
[The figures on this plate are all from untouched negatives.!]
Fig. 1. Portion of spinal nerve of stage 34. (108'28 35.) l.vag., lateralis vagi; my. myotome;
n. notochord ; n.t. nerve trunk ; n.s. nuclei of protoplasmic sheath ; d.r. dorsal root ; v.r. ventral root.
Fig. 2. Portion of spinal nerve trunk of stage 29.* (93 C. 1435.) my. myotome; n. notochord;
n.t. motor nerve trunk ; p.s. protoplasmic sheath containing yolk granules (black), and large nuclei rich in
chromatin.
* Op. cit., p. 238.
t Much of the minute detail has unfortunately disappeared in the mechanical printing of these figures. I shall be
glad therefore to show any specialists who are interested sun prints from the same negatives, in which the full detail
is brought out.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 7) 21
128 EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES.
Fig. 3. Part of transverse section at stage 27. (69 C. 1064.) Showing whole length of motor nerve
trunk, my. myotome ; n. notochord ; n.t. motor nerve trunk ; p.s. protoplasmic sheath now in form of a
mass of richly yolked mesenchymatous protoplasm aggregated round the nerve trunk near its outer end.
The greater part of the nerve trunk is still naked. At its outer end the nerve trunk passes out into conically
arranged strands of protoplasm forming the inner ends of the muscle cells.
Fig. 4. Part of transverse section at stage 25. (79 A. 933.) my. myotome ; n. notochord ; n.t. motor
nerve trunk already faintly fibrillated passing between spinal cord (s.c.) and myotome. The mesenchyme
(me.) has not yet begun to concentrate round the nerve trunk.
Fig. 5. Part of transverse section at stage 24. (73 G. 986.) my. myotome; n. notochord; n.t. motor
nerve trunk, at this time composed of granular protoplasm, and naked.
Fig. 6. Similar section to last, but taken from rather less advanced specimen. (Stage 24 ; 73 F. 688.)
my. myotome ; n.t. motor nerve trunk — a naked protoplasmic bridge connecting myotome and spinal
cord (s.c).
Plate II.
[All figures are camera drawings of single sections. Figs. 7, 8 and 9 represent the same sections as are
photographed in figs. 2, 3 and 4, with the additional detail visible under Zeiss' 3 mm. apochromatic
homogeneous immersion objective.]
Fig, 7 ( = fig. 2). The motor nerve trunk (n.t.) is seen within its thick protoplasmic sheath (p.s.).
Fig. 8 ( = fig. 3). The motor trunk is now naked except for the large mass of yolkladen
mesenchymatous protoplasm (p.s.) which has concentrated round it towards its outer end. The continuity of
nerve trunk with protoplasm of muscle cell is seen.
Plate III.
Fig. 9 ( = fig. 4). The nerve trunk is seen to be already fibrillar in structure. Mesenchyme (me.)
has penetrated in between myotome and spinal cord, but has not yet begun to concentrate round the nerve
trunk to form its sheath.
Fig. 10. Stage 24. The nerve has been torn away from the inner surface of the myotome so that its
expanded outer end is seen.
Fig. 11. Part of transverse section, stage 30. (93 B. 2844.) The two walls of the myotome are seen —
the outer onelayered, the inner composed of a layer of myoblasts in which are seen the first contractile
fibrils (c.f.).
Fig. 12. Part of transverse section, stage 31. (103 D. 2463.) The outer wall of the myotome is beginning
to show more than one layer of nuclei. In the myoblasts of the inner wall the contractile fibrils (c.f.) have
greatly increased in number, most of the protoplasm of the inner half of the cell being converted into fibrils.
Plate IV.
Fig. 12a. Part of longitudinal horizontal section at stage 31 (103 C. 822), showing a single myotome.
i.w. Inner wall cell ; c.f. contractile fibrils ; g. glycogencontaining outer part of cell ; nu. nuclei of inner
wall cell ; y. yolk granules ; o.w. outer wall of myotome, two of the nuclei undergoing mitosis : me.
mesenchyme nuclei of septum between myotomes.
Fig. 13. Part of transverse section at stage 31 + . (106 C. 1573.) The outer wall of the myotome is
now several layers thick, and the cells of this wall have also developed contractile fibrils (c.f,).
Fig. 13a. Longitudinal horizontal section through a myotome of stage 31 + . (106 A. 1811.) The
nuclei of the inner wall myoblast are seen to have considerably increased in numbers. The contractile fibres
of the outer wall (c.f.) are visible.
Fig. 14. A nucleus of one of the inner wall myoblasts during mitosis, showing the celllike demarcation
of the protoplasm immediately surrounding it.
v r: Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa.— Plate I.
Sol. Edin.
my
FIG. I.
FIG 4
FIG 2
FIG 5.
Wm {
\
/
trunk.
I protopla
re II.
fk on i on
ions. Figs. 7 ; e same sections as are
3 mm. apochromatic
Co
^ ^
^?°°
f^^/p
A*
Q
O
o
I
e an
s\/' X' ( . . <tf) the nreQ^niV^s oeaiiinJAur
 nifccontractile fibrils\(c v
S OH
„ lewTurns' iiuc
•'P^iwng mitf
• '■ ; ng a single myotome.
c on
i r: Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa.— Plate I.
ySoc. Edin. Vol. XLI.
B00 hB&
■s
Vans. Roy. Soc.Edin 1
KERR^ EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS
AND MYOTOMES IN LEPIDO SIREN PARADOXA. PLATE 2.
Vol.XLI
Fig. 7.
Pig.8.
E .Wilson , Cambridg e .
Roy. Soc.Edm T
KERR: EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS
AND MYOTOMES IN LEP1DOSIREN PARADOXA. PLAT
XL I.
E Wilson, Cambridge
y
Soc.Edm 1
Vol.XLI.
KERR: EARLY DEVELOPMENT OE MOTOR NERVE TRUN]
AND MYOTOMES IN LEPIDO SIREN RARADOXA . PLATE 4.
Figs. 12 A, 13, 13A.
I I I I I I I I I I I I
c.f! o.w.
13A.
■~.yv
E . Wilson, Canibridg e .
Trans. Roy. Soc. Edin.
Vol. XLI.
Prof. J. Graham Kerr on "Some points in the Early Development of Motor Nerve Trunks and Myotomes
in Lepidosiren paradoxa." — Plate V.
DESCRIPTION OF TEXT FIGURES AH.
Camera outlines of portions of transverse sections of the trunk region of young Lepidosirens at various
stages, to show the topographical relations of the myotomes. Zeiss Obj. A, Oc. 2. i.w., Myoblasts of inner
wall of myotome ; /. vag., lateral branch of vagus.
Fig. A. — Stage 16.
Fig. a— Stage 21.
Fig. C. — Stage 24.
Fig. D. — Stage 27.
Fig. E. — Stage 31.
Fig. ¥. — Stage 31 +
Trans. Roy. Soc. Edin.
Vol. XLI.
Prof. J. Graham Kerr on "Some points in the Early Development of Motor Nerve Trunks and Myotomes
in Lepidosiren paradoxa."— Plate VI.
1 ' £glr* •• :  ft
■ W°C   •* .. „"•> ZZSjfU fj f,
sv
I
Fig. G.— Stage 34.
;...
Pig. H. — Stage 36 + .
( 129 )
VIII. — An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate.
By John Dougall, M.A. Communicated by Professor G. A. Gibson.
(MS. received February 4, 1904, Read March 21, 1904, Issued separately August 5, 1904.)
The following paper contains a purely analytical discussion of the problem of the
deformation of an isotropic elastic plate under given forces. The problem is an unusually
interesting one. It was the first to be attacked (by Lame and Clapeyron in 1828) after
the establishment of the general equations by Navier. The solution of the problem of
normal traction given by these authors, when reduced to its simplest form, involves
double integrals of simple harmonic functions of the coordinates. The integrals are of
complicated form, and practically impossible to interpret, a fact which, without doubt,
has had much to do with the neglect of the problem in later times, and the almost com
plete absence of attempts to establish the approximate theory on the basis of an exact
solution. An even more serious defect of Lame and Clapeyron's solution is that the
integrals, as they stand, do not converge. A flaw of this sort has often been treated
lightly by physical writers, the nonconvergence of an integral being regarded as due to
the inclusion of an infinite but unimportant constant. In the present case, however,
the infinite terms are not constant, but functions of the coordinates, and the modifica
tions necessary to secure convergence, so far from being unimportant, lead directly to
the most significant terms of the solution.
The next writer to deal with the exact problem was Sir W. Thomson, who, at the end
of the memoir in which he solved the problem of a spherical shell, indicated the form
which the solution would take in the limiting case of a plate. His method, if carried
out, would lead to integrals of the same form as Lame's.
Solutions of special problems have been given by other writers. Prof. Lamb has
worked out the solution for an infinite solid subjected to normal pressure proportional
to cos kx, and verified in this particular case some of the results of the approximate
theory of thin plates (Proc. Lond. Math. Soc, vol. xxi., 188990).
The history of the approximate theory is well known and easily accessible. It will
be sufficient here to refer to —
(i) Todhunter and Pearson's History of the Elasticity and Strength of Materials.
(ii) Clebsch's standard treatise, Theorie de Velasticite des corps solides, as trans
lated by Saint Venant.; in particular, Part I. chap, iii., and Saint Venant's brilliant note
on § 73.
(iii) Prof. Love's Treatise on the Theory of Elasticity, 1892, — especially the
historical introductions to both volumes.
The various forms of the approximate theory rest partly upon the general equations
of equilibrium, partly upon auxiliary hypotheses or physical principles. These
principles are recognised as contained in the general equations, but on account of the
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 22
130 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
analytical difficulties in the way of deducing them rigorously, they are either simply
assumed, or else supported by reasoning plausible rather than demonstrative.
In the following pages the problem is treated as a purely mathematical one, and the
approximate theory for a finite plate deduced from an exact. solution for an infinite plate.
The main features of the method are —
(i) The use of Bessel functions in place of the simple harmonic functions of previous
writers. Only the symmetrical forms, or functions of order zero, are required.
(ii) Transformation of the definite integrals, in terms of which the solutions are in
the first place obtained, into series, by means of Cauchy's theory of contour integration
and residues. The series involve Bessel functions of the second kind with complex argu
ment, and are so highly convergent that the principal features of the strain represented
by the solution can be made out with the utmost ease. (The transformations belong
to a class discussed systematically, apparently for the first time, in a paper " On the
Determination of Green's Function by means of Cylindrical and Spherical Harmonics,"
Proc. Eclin. Math. Soc, vol. xviii.)
(iii) Detailed solutions of the problems of internal force with vanishing face traction.
The usual method of dealing with a general problem in Elasticity is to find a particular
solution for the bodily force, and then to treat the problem of surface tractions
completely. This is theoretically sufficient, but leaves the result in a complicated form,
which in the present case must be simplified before practical applications can be made.
(iv) Use of Betti's Theorem (Love, Elasticity, vol. i. § 140) to develop a method
analogous to the method of Green's function in the Theory of the Potential, by which the
properties of the solution for a finite plate can be deduced from the infinite plate solu
tion. (Cf. Proc. Eclin. Math. Soc, vol. xvi., "On a general Method of Solving the
Equations of Elasticity.")
The results of the ordinary theory are fully confirmed, and extended in various direc
tions. The infinite solid solution gives, of course, an exact particular solution for
internal force and traction on the plane faces of a finite plate. At the head of the solu
tion appear the terms given by the approximate theory. In the case of flexure, the
equation of Lagrange is obtained to a second approximation.
The problem of a finite plate under given edge tractions cannot be completely solved,
but exact solutions are given of certain problems relating to a circular plate. For a thin
plate, with edge of any shape, the conditions satisfied at the edge by the principal terms
of the exact solution are found to a degree of approximation beyond the reach of any
theory which rests merely on the " principle of the elastic equivalence of statically equi
pollent loads." For example, the celebrated boundary conditions given by Kiruhhoff, in
correction of Poisson, are verified, and extended by the inclusion of terms of higher order.
In conclusion, it may be mentioned that the methods given here are equally appli
cable to the problem of the vibrations of a plate, and to the problems of the equilibrium
and vibration of a finite circular cylinder, or of an open spherical shell. Some account
of these applications I hope to publish shortly.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
131
TABLE OF CONTEN T S.
(i), 0)
<*)•
PAGE
Introductory Analysis 132
(a) to (</). Evaluation of definite integrals involving
the Bessel functions. Degenerate cases . 132
(h) Potential functions derived by integration from
the pointsource potential. Definite integral
expressions for these 135
Solution of the problem of flow between
two infinite parallel planes. Main stream
and local currents 136
Curvature. Differentiation as to arc and
normal ....... 139
8 1. Equations of equilibrium. Form of solution
for a plate free from bodily force . . 140
§ 2. Force applied at a single point of an infinite
solid 142
§ 3. Solution of the problem of normal traction
for an infinite plate ..... 143
§ 4. Flexural and extensional components of the
strain. Disadvantages of the solution in
definite integrals ..... 145
§ 5. Transformation of the definite integrals into
series, by means of Cauchy's Theorem of
contour integration 147
§ 6. Types of the particular solutions composing
the general solution 1 48
§ 7. Position of the zeroes of the functions sinh
C±C 150
§ 8. Approximate forms of the »t th terms of the
infinite series, when n is large . . . 151
§ 9. The solution for arbitrary normal traction.
Questions for discussion . . . . 151
§ ] 0. Detailed solution of a special case. Term by
term differentiations . . . . .152
§ 11. The same special problem. Summation of
two infinite series 154
Final form of the same special solution . . 155
Order of magnitude of the various parts of
the solution when the thickness of the
plate is small . . . . . .156
Methods and results of the sjJecial case ex
tended to the general problem of arbitrary
normal traction 157
§ 15. Independent symbolical solution of the
general problem ...... 160
§ 16. The problem of tangential face traction.
Solution for an element of traction . . 161
§ 17. Composition of the solution . . . .163
§ 18. General solution. Comparison with the
solution for normal traction . . .165
§ 19. Normal force applied at a single internal
point. Solution in definite integrals . 165
§ 20. The same solution in series . ... . 169
§ 12.
§13.
§14.
§ 21. Solution of a special problem of internal
areal normal force .....
§ 22. Approximate forms of displacements and
stresses in the general case ....
§ 23. Normal force of constant intensity throughout
the thickness ......
§ 24. Internal force parallel to the faces
§ 25. Solution .......
§ 26. Approximate results. Lagrange's equation
for flexure to a second approximation
§ 27. Extensional strain. Differential equations of
the principal mode .....
§ 28. Approximate values of the stresses across a
plane parallel to the faces ....
§ 29. Transmission of force to a distance. Expan
sions in polar coordinates ....
§ 30. Types of deformation conveying a given
resultant stress
Conditions for the existence of a solution with
finite potential energy. Elastic equivalence
of statically equipollent loads
Betti's reciprocal theorem. Verification of
preceding solutions
Finite plate under edge tractions. Form of
the solution deduced by means of Betti's
Theorem
The same by another method
General solution for an infinite solid under
any forces .......
§ 36. Betti's Theorem and the problem of given
edge tractions
§ 37. Exact solutions of special problems for a
circular plate. Problem 1 — symmetrical
transverse displacement ....
§ 38. Problem 2 — normal displacement and nor
mal shearing stress given. The Fourier
and other methods of obtaining such
solutions .......
§39. Problem 3 — permanent modes due to sym
metrical edge tractions ....
§ 40. Expansions of arbitrary functions .
§41. The problem of given edge tractions for a
thin plate .......
§ 42. Extinsioial strain .....
§ 43. The Green's function method for the per
manent mode
§§ 44, 45. Flexural strain. Solutions to first and
second approximation . . . 218,
§ 46. Flexural strain. The Green's function
method. Kirchhoff's boundary conditions
to a second approximation
Addition to Paper
§31.
§32.
§33.
§34.
§35.
PAGE
170
173
175
176
178
180
181
182
183
185
187
189
192
195
196
197
198
201
205
206
208
208
213
220
224
228
1 32
y\\\ JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Introductory Analysis.
(a) The Bessel function J is defined by the series
Z m / Z 2 2 4
JmOO = 2 m nm\ ~ 2 • 2m + 2 + 2 • 4 • 2m + 2 ■ 2m + 4
For the function of the second kind we take as definition
2 sin mn
This makes G,„z an analytic function of m , the value of which, when m is a real
integer, is
G m ( Z ) = (log2 r + ).J m ( 2 )Y m (,)
where Y m (z) is Neumann's function.
In this case, therefore, G m z = — log zJ TO z + a uniform function of z.
In the following pages we are concerned chiefly with the function of order zero
^r+T^p+KP
)iog*(iJ + ^ 2  ■ ■ . )£j + (i+^L .
When mod z is very large, while the phase (argument) of z lies between — ir/2 and Stt/2,
then approximately
G m z = e —e<>+4 }
2z
Similarly, when the phase of z is between and 7r (excluding those values)
(6) If .'• , y , z and p , a> , z are the rectangular and cylindrical coordinates of a point
in space, so that x = p cos w , ?/ = p sin w , then the most important property of the Bessel
Functions is that each of the eight functions
(e KZ or e~ KZ ) (J m Kp or G m Kp) (cos wito or sin mot)
satisfies Laplace's equation, or in other words is a potential function.
Hence
(y 2 + k 2 ) • (J m /cp or G m Kp) (cos mu> or sin mm) = .
Further, if
= p* + p * — Ipp COS (OJ  CD )
then
(V 2 + *'') • (J «R or G kR) = .
Let now I = / / G kK/(:c', y')dx'dy\ the integral being taken over a finite area A.
Then ( v 2 + « 2 )I = , if (as , y) is without,
THE EQUILIBRIUM OP AN ISOTROPIC ELASTIC PLATE. 133
but ( v 2 + k 2 )I =  27r/(a; , y) , if (x,y) is within this area; as easily follows from the
theorem
V 2 f f log Rf(x', y')dxd,f = 27rf(x , y) .
The differential equation satisfied by I, together with the conditions that I and its
first derivatives dl/dx , dljdy are continuous throughout, define the value of the integral
completely, and in many cases make its evaluation easy.
(c) For example, takef(x , y) = J m /3p cos m&> , with m an integer, so that
I = l I G kR J m f3p' cos mm' p dp dm
and suppose the area of integration to be a circle of radius a, with centre at the origin.
For convenience in the proof, let the imaginary part of~/c be positive.
Then I = ^ J m /3p cos m<o + AJ m <p cos mm , when p<a ,
p z — k
= BG„,Kp cos mm , , when p>a
A , B are determined from the conditions that I and dl/dp are continuous at p = a.
Thus we find
I = „ 2 ~^" 2 J m /3p cos m <» + 02 ^ 2 J "' fcp cos mm ( Ka ^ r m xaJ m /3a  G m Kaf3aJ J fta) ; (p< a)
= n 2 _ 2 ^m K P cos mw(KaJ m 'KaJ m /3a  J m Ka^aJ„'fSa) ; (p> a )
By the principle of continuation in the Theory of Functions, the result is true
whatever be the phase of k. But when the phase of k is diminished by 2n,
G m (/<c) is increased by 27riJ m (i<c)
and
G m '(Kc) by 27rz'J m '(Kc) ;
hence, equating the corresponding changes in I and its value, we obtain
I I J /cR J m Bp cos mid p dp dm
J J
2tt
= 02 _ 2 ^m K P cos m(ji(KaJ m 'KaJ m /3a ~ J„,Ka^aJ m 'fSa) .
From this again it easily follows that in I and its value we may replace the G functions
by the Y functions.
(d) We have
Y /cp = log KpJ Kp + £* 2 p 2
= log kJ k P + log p(l  kV . . . ) + kV . . .
Thus log/c J Kp  Y Kp is an integral function of k, in which
coefficient of k° is — log p ,
and coefficient of k 2 is ^p 2 log p  \p 2 .
134 .MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
The functions log p and ^p 2 \ogp — ^p 2 are thus in a sense degenerate forms of the Bessel
Functions, and any theorem relating to the G or Y functions will yield a corresponding
theorem in these.
Thus by equating coefficients of k 2 in the equation
( v 2 + « 2 )(logKJ K /3 Y K P ) =
we obtain
V 2 logp =
V 2 (V log p{p 2 ) = log p
and therefore
V *(ip 2 logpip 2 ) = 0.
We deduce at once
V 2 jf(W log R  \W)f(x, y')dx'dy' = jf log Ef(x, y')dx'dy'
and v4 // (iR2 l0g R " * R2)/(a; '' y') dx ' d y' = v '// lo § M<y')dx'dy'
= 2tt/(* , y) .
(e) Again, from the addition theorem
Y /cR = Y KpJ Kp'+ 2 y ,Y m KpJ m Kp cos m(u  oj") ; (p>p)
m=l
we deduce
logR = logp  2,— f ) cos m(o>  a)') ; p>p'
tri"Ap/
and
iR2 log R  JR* = (Jp2 log p  p2) + ip'2 log p
+  (P  2p log p)^  _  cos (oi  o>')
( /') In the same way, from the results of (o), we may deduce the value of the
integral
^o = I / (4R 2 1°§ K ~ iR 2 )Jm/3p' cos moi p dp dm' .
The form of the result varies in the cases m = 0, m=l , m>l.
m=0 :
Io = ^ofr + J,{ {^ ~ P j)(loga/3aJ '/3a J /3a)
+ (l ] og a\l3aJ '(3a + ? f 2 log a  1 JJ /3a 1 , p< a
7n=l :
$ { ( 4 ^ l0§ P ~ 4 p2 )( ~ ^ aJ o'^ a ) + log p ( 7p  J^o'/ 30 + \ J oP a ) } . P > a •
+ ?Vl + 2 log a) J^a + 2?(l  2 log aWr/jSa \ , P <a
.Jac*. j^£ + l p2 _ l^iogp^a^aJ^a) + '(/taJ/j8a  SJ^a) } , p>a.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 135
m >1 :
2tt t n 2tt p m COS mm ( ( 1 p 2 Yo t'/d , T o^
I = _J m /3pcosma,  _L^_ (^  jE^lfaJ.pa + rnJ^
+ 7 — ^2\ftaJ m 'f3a + m  2 J m /3a ) > , p<a
The corresponding integrals with log R in place of ^R 2 log R  ^R 2 may be obtained at
once by taking y 2 0I> * ne above.
Also, all through we may write log (R/c), log (p/c), log (a/c), instead of log R, log p,
log a, this amounting merely to a change in the unit of length.
(g) By equating coefficients of like powers of /3 in the results of (c) , (/) we can
obtain
I I G /cRp' m+2 "cos moi'p'dp'dm
J o J o
and T P^R 2 log R  iR 2 )p""+ 2 » cos mm'p'dp'da.
In the case when to = n =
// Q a KRp'dp'dw' = — — J Kp«aG 'Ka , p <a
y K 2 K J
= — — „ G Kp/<aJ 'Ka , p>a .
Il llX \ R2 l0g R ~ I B ^)p' d P d<0 ' = § { P * + 4p2a2(2 lo S " ~ J ) + " 4 ( 4 !og «  5) 1 , p < a
= ( jP 2 logP  p 2 J7ra 2 + logp^ , P >a.
These results and those of (f) may easily be verified, or obtained, from the values
dl d
of v 4 of the integral, with the conditions that I , r , y 2 I > ^TV 2 ! are continuous at p — a.
(h) In certain problems a class of potential functions occurs, which may be deduced
from the fundamental potential l/r, where r 2 — x 2 + y 2 + z 2 , by successive integration
with respect to z.
Writing
u x = log (r + z)
u 2 = z log (r + z)  r
u s = i( z2 ~ IP 2 ) lo g (r + z)~ \rz + £p 2
we may easily verify that u lt u 2 , u s are potential functions, and that
du s
~dz
= u.,
du.y
dz
= «i
du x
1
~dz
r
136 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
These ^integrals of 1/r may be expressed in the form of definite integrals involving the
Bessel function J, analogous to the integral forms for r~ l and its zderivatives,
r
d
•~ l = i e" z J Kpc?K
f
Y r~ l = I (  K)e ~ KZ J KpdK , etc., where z > ,
ran
We may notice that the value of I e'^^pch follows at once from the remark that
it is a potential symmetrical about the axis of z, and taking on that axis the value
f° 1
I e~ KZ (h =  . We may use this idea to express w : , u 2 , u 3 in similar form.
z
For we have
j"( e  K z_ e  Ke yjL = log*,
and
/;
\dK . z
qkz _ i + KZ e KC — ^ = zlog z
J K Z ° C
1 „ „ \d K 1 „. 2 3
f a (e*l + KZ  ^¥r«^=  J Z 2 1og^+ ~z
by integration with respect to z from to z.
Hence
r + z
=  l0 §'2c
I (e" z J Kp l + Kze Ke j
(Ik , r + z
because in each case the functions equated are symmetrical potentials, taking the same
value on the axis of symmetry.
By putting z = in the first and third of these we obtain two integrals, of great
importance in the following analysis,
f o (j « P  e «)^= io g £
There is no difficulty in generalising the above results, but those given are all that we
shall require.
(i) With a view to indicating the broad lines of the treatment ' of the elastic
problem given in the succeeding pages, a discussion on similar lines may be given here
of a simple problem in potential, in which the attention is not distracted from the
principles of the method by any complexity in the calculations.
The problem is to find the flow from a source situated between two parallel planes
z = db h, under the condition that there is no flow across these planes.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 137
We require a potential V, becoming infinite as l/r &t (x' , y f , z'), but with no
dV ■ '
other singularity at a finite distance, and such that 77 = when z = dzh.
then r~ l = / n~ K<z ~ z '\J KRdK , when z>z
J
= I e K ' ^ ~ ^ ' , J^HdK , when z<z.
J u
— r~~' = I ( K)e~ K{ '"J K'RdK, when 2>x'
= I Ke" ,,, * ,, .T icKd/c , when g<z'. I
.'
We therefore begin by finding a potential
Y x = (A cosh kz + B sinh kz)J kR
giving
We obtain
 d j = Ke K2  Z ''J KR on z = 7i
=  k^ u  j, 'J a kR on z= h.
A sinh k/< 4 B cosh k// = e^ Kl ' i ~ z ' >
A sinh k/i  B cosh nk = e~ K(h+x ''
cosh kz
A = e" A
B = e~ Kh
sinh k7'
sinh kz
cosh k/>
v  <cj/ cosn k;2 cos h *z' sinh kz sinh kz'\ t
y \~ e [ IrCTS + TTTi— I J«V
V sinh kIi cosh /c/t
cR
If this could be integrated with respect to k from to 00 , we should have a
potential just balancing at the boundary the flow from the source.
But V x becomes infinite as l/ich at /c = , and the integration cannot be performed.
We may, however, subtract from Vj the (constant) potential e' KC JKli , where c is an
arbitrary positive quantity. This makes integration possible, without introducing any
flow across the boundary.
A solution of the problem is then
L f[ , /cosh kz cosh kz' sinli k2 sinh kz'\  „ e~ KC
V =  +
/"°° { , /cosh kz cosh kz sinh k2 sinh kz\ „ e~ KC 
J e Khl ^_^ + \j kR _ V ,
J „ ( \ sinh k/i cosli /c/i y ° K/i )
But this form of solution, while theoretically complete, is of little value because of the
difficulty of interpretation. For example, it gives no indication of what on physical
grounds we should expect to be the chief feature of the phenomenon, namely, the
practically twodimensional character of the flow at a moderate distance from the
source.
The transformation to which we proceed brings this out as luminously as possible.
First, it is convenient to separate Y 1 into its odd and even parts in *, as is easily
done by writing cosh k/i — sinh <h for e~ Kh .
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 23
138 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Thus
/ cosh ah ■> i > sinh kIi ■ , • 1 At o
^i = (  cosh kzz' H cosh kz cosh kz  sinh kz sin Iikz )J kR .
1 \ sinh kH cosh k/i /
Next we replace the term 1/r in V by the equivalent integral
/" e^'^JoKEdK .
Hence
V = / i s ; n h kzz 4 K cosh kz cosh kz' — sinh kz sinh kz' )J kR  — — > i
Jo (V sinh x/i cosli k/i J kIi )
the upper or lower sign being taken in the ambiguous term according as z is > or < z'.
When R > , this integral can be separated into the two
/'% ,../ , cosh x/i , , , sinh k/i . , . , , 1 \,
J„kR + sinh kz  z + ^j — 7 cosh kz cosh kz —  sinh kz sinh kz  Wk
J u \ sinh K/i cosh k/i khz
1 R
The value of the latter inteoral we have found to be — = log tt •
The former integral is of the form I J,/RF(/c)<i/c, where F(*r) is an odd function of *c,
vanishing for < = 0. It may be expressed as a complex integral
— ; I G kRF(k)cZk,
the path being from west to east along the whole of the real axis in the k plane, for
G (kR)  G (k^R) = 7rt'J KR .
Now, from the original form of V 1? and the integral forms of l/r, it is obvious that
F(/c) vanishes at infinity in the eastern half of the k plane ; being odd in k it must
vanish likewise in the western half.
Hence by Cauchy's Theorem, the integral — ./G kRF(>c)o?/c is equal to twice the sum
of the residues of the function G kRF(/c) at its poles in the upper half of the < plane,
and
v =  Nl
2 "^ nirz nirz inirK
V — jL^ cos — j— cos .  G
II , l = l /i li " /i
1 Y'wB
2/ A
+ lZ siu ( n + lTi sm ( n + lTi G ( n +
(j) The solution indicates (i) a main current in two dimensions, defined by the
1
h
■I T>
potential — j log 9 , and (ii) an infinite series of local currents in three dimensions,
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 139
practically insensible when the distance from the source is a moderate multiple of the
thickness of the plate. In the following pages we shall deduce analogous solutions for
' sources of strain ' of the different types which may exist in an elastic solid, and develop
these solutions in various directions. The corresponding development of the present
solution is extremely easy, but would carry us too far. We merely mention that the
' main current ' in the hydrodynamical problems corresponds to the ' principal modes
of strain,' the determination of which is the object of the theory of thin plates. But
there is one important distinction in the two cases. In the flow problems the
exact conditions defining the ' main current ' can always be found, and are indeed
obvious ; on the other hand, the analogous conditions in the strain problems can only
be found by approximation.
(k) The following conventions seem to be very generally adopted, but to prevent
any risk of ambiguity they may be stated explicitly here. Consider any continuous
plane area A bounded externally by a closed curve C , and internally by one or more
closed curves Cj , C 2 , etc. At any point E of a bounding curve let Ex , E// be drawn in
the directions of the rectangular axes of coordinates. Let Ex , Ei/ be turned through an
angle e, which will be taken as positive when the rotation is counterclockwise, until
they coincide with E£ , E>? , the direction of E£ being that of the normal at E when
drawn from within A towards the boundary. E£ , E>; will be taken to be the positive
directions of the normal and tangent at E, and \£f(x, y) be any function given within
A, / and ~ will be used to denote the rates of variation of f per unit length in these
CL7Z (a/o
positive directions.
The curvature at E is ^ and is denoted by I /p. p is therefore positive when, in
order to reach the centre of curvature from E, we have to proceed into the area A.
If we suppose the figure traced on level ground, a person proceeding along the
boundary in the positive direction will have the area on his left, and the curvature
will be positive when he is rotating about the vertical in the counterclockwise sense.
The following formulae relating to differentiation along the arc and normal will be
much used in the later sections of the paper. Suppose the axes of x and y to coincide
with the positive normal and tangent at a point of the bounding curve. At a
neighbouring point E (x, y) on this curve
df df , . dr
f = COS e — + Sin e z ,
dn dx dy '
df ■ df , df I
j = — sin e ~ + cos e — [
ds dx dy ,
By putting x, y, e equal to zero, we have at
df = df 4f = df
dn dx; ds dy ^ '
140 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Differentiate the first of equations (i) with respect to s. Thus
( d £) = C0S€ < L( < V)
\dnj ~ ' ds \dx)
d_ (df
ds
d (df\ . <h df de df
+ sin € j [± I  sin € _ f + cos c — /
as \dyj as dx ds dy
<]1
dx 2
3'
dxdy
and at 0, **&+*&
ds dn dx dy ds dy
or
d 2 f _ d df 1 df
dxdy ds dn p ds
Similarly from the second of (i),
ds 2 dy 2 p dx
or c ^ f = c El+ lc ^
dy' 2 ds 2 p dn
(iii)
(iv)
Thus the values at of j. /, ^A, t4 are known when /'and / are given along
dx dy' dx ay dy 1 J dn ° &
the boundary.
~d~ 2 + ct 2 or v!/* Dem g an invariant for all systems of rectangular axes, we may also
conveniently take
<iy = 2f _i_tf_c£f
i/.r p dn ds
(v)
1. Equations of equilibrium. Form of solution for a plate free from bodily force.
The equations of equilibrium of a homogeneous isotropic elastic solid are of the
form
xx d xi/
'lx dy
d
i
d xij
lx
d 7:.
dx
dz
+ X =
d yy ^'dyz + y = Q
dx dy dz
d yz d & v
+  — +  + Z
dy dz
(1)
xx, yy, zz,
where X, Y, Z are the components of the bodily force per unit volume, and
xy, xz, yz are the components of stress, these being given in terms of the displacements
u, v, iv by the equations
xx — AA + 2/x
yy = AA + 2/x
\ a « dw
AA + 2/x—
where
~ (dw dv
yz = *\Ty + &
^ fdu dw\
,z = \dz + dx)
— (dv du\
dz~ ' xy = ^{dx + dy)
du
dx
b:
\
(2)
A =
an
du dw
fx dy dz
THE EQUILIBRIUM OE AN ISOTROPIC ELASTIC PLATE
141
In terms of the displacements, the equations of equilibrium are therefore
MV 2 M + (A + M ), A + X =
dx
HV 2 v + (* + A*)^r + Y = .
dy
yu.VW + (A + /x) + Z
dz
(3)
(i)
(»)
When the bodily force is null, or X = Y = Z = 0, the following forms are easily shown to
satisfy equations (3),
"dy\
dd\
dz
dO
dy
06
dz
\ + % l xd<t> + il? d i <j> \
A + /A dx dz dx
A + 3/A c?c/> 2 rf 2 <£
k + jx dy dz dy
(iii) m =
w =
A + 3/A <7<£ 9 r7 2 (/>
A + p dz
where ^, 0, (p are potential functions, so that
vV = °> v a # = 0, v 2 <£ = .
These solutions have been used by Boussinesq in his treatment of the problem of a
solid bounded by a single plane z = 0. They are equally effective when the boundary
consists of two parallel zplanes. Thus, as will explicitly appear in the sequel, and as
might be proved at once, any solution of (3), with X = Y = Z = 0, in the space between
the planes z= ± h, can be expressed in the form
n dil/ dO dd, , d~4> \
u = 2 r + — + af + 2z— ?
dy dx dx dz dx ,
v = 2 d A + d l +a ^ + 2z^
dx dy dy dz dy
w =
dO
dz
dd> n d 2 d>
i^ + 2z— £
dz dz 2 J
(4)
Here, and throughout the paper, the symbol a is used to denote the fraction (X + 3m)/
(a + m).
With these values of u, v, iv the stresses across a zplane, viz., zx, zy, 7z, are given by
zx
2/JL
2/,
<m
. .  ~^r + 2z
d 2 4>
d 3 <f> )
d 2 ty
dy dz "^ dx dz ' dx dz
dV ^ d*6 ^ d 2 <f>
dx dz' 2
dx dz dy dz ' dy dz
d 2 6 d 2 <j>
~dz 2
+ 2z
dy dz 2
+ 2z^±
dz 2 dz s
(5)
142
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
2. Force applied at a single point.
Returning to the solutions (i), (ii), (iii), we note that (i) and (ii) contribute nothing
to the dilatation A, and (ii), (iii) nothing to the zrotation a> s = ^(^  J) ■
These properties can be used to resolve any given displacement into its ^, 0, (p
components, the bodily force being null.
An example of fundamental importance is the displacement in an infinite solid due
to a single force applied at a given point. Thus for a unit force applied at the origin
in the direction of the axis of z we have
u
\
vrz
V
=..y z
,.s
w
2 2 a
= S + 
► each multiplied by —
1 \+jl _
where r 2 = x 2 + y 2 +
have
8nfi A. + 2/a 47r / a(a + I )
or say, for a Z force of 4717* (a+ 1) units applied at (x\ y' , z') we
dr~ l
U = (z'  Z)
v ={z'z)
dx
dy
dr 1
v= (z'z)% +ari
dz
(6)
r x being written for l/r, where r is the distance from (x, y, z) to (x', y', z').
These give
But in (4)
A= (al)_ ; <o, = 0.
dz
A = 2(la)ft;"3 = ft
dz dz
Hence we take ^ = 0, and choose (p so that
d<f>
dz"
2 ?
Now the functions log [r+z — z') and  log (r zz') are both potentials having r' 1 for
zderivative ; the former is without singular point in the region z > z', the latter in the
region z < 2'. We may without confusion use a single symbol to denote either function
indifferently, and define
dz~
We may therefore take
r = log (r + z z) when z > z \
= log (r z + z) when z<z \
(7)
1_ d l r~ l
J ~dz~'
Comparison of the displacement w in (4) and (6) gives now
d$ ,dr
= z — + ~ r 1 .
dz dz 2
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 143
For a Z force of 4^(01+ 1) units at (x', y f , z') we have therefore
t/r=0
, _! a d V
2 cfcr 1
1 d" 1 /" 1
^~ 2" "dJF 1 "
(8)
It is easy to verify that these values of >!/, 0, <£ substituted in (4) do actually reproduce
equations (6).
Similarly for an X force of 47r / u(a+ l) units at (x\ y', z') we find
_a+l c2 d 2 r l
*~ ~2~~ Ty dz~ 2
Z T
1 rf dr'r 1
4> =
^ 2 dz dz~ 2
(9)
Here — 2  denotes a potential function having j 7=r for z derivative, and is defined
by the equations
— — = (z z) log (r 4 z  z)  r when z>z [ /, ~v
= — "(2  z') log (r  z 4 z')  r when z<z' J
It may be observed that the necessity for dealing separately with the two regions
z>z' and z<z' in these cases is not inconsistent with the theorem of (4), which refers
only to a displacement free from singularity in the space considered.
3. Solution of the problem of normal traction.
Coming now to the problems relating to a solid bounded by the two parallel planes
z — h and % =  h, we begin with the simplest of these, and seek a solution of the
equations of equilibrium giving
X — Y = Z = throughout the hody ;
the normal stress
w=f(*,y) on z = h,
= on z =  h ;
the tangential stresses zx, zy = on both faces z = ± h. The arbitrary function f(x, y) ,
which we shall suppose to vanish at all points without a given finite area A, is expressed
in a form amenable to analytical treatment in the familiar theorem
Limit f ( */(*', y')dx'dy _« />„ a ni x
the integral being taken over the area A.
(If we imagine the plane z = to be covered with attracting matter of surface density
f{ x > 2/)> tnen the theorem expresses the wellknown relation between the density at
144
MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF
(.'•, //, 0) and the limiting value of the normal attraction at (x, ?/, e) as e approaches
zero. )
As a preliminary to the general problem, we take therefore the special case in which
« on z = h is equal to e/{(x — x'f + (y — ;/) 2 + e 2 } , or, in the form of a definite integral,
I c^kJi/kR)^,
R 2 = {xxf + (// 2/')2.
where
Making a further reduction, we begin by taking, in place of this integral, simply the
function kJ /cB,.
The function ^ is not required, and <p, are of the forms
<£ = (C\ sink kz + C. 2 cosli kz) J kR i
6 = (C 3 sink kz + C 4 cosh kz).T kR J
In accordance with (5) these satisfy the conditions
(16 , dd> , n '7 2 (& A . ,
— + — ' + 2z — i =0 on z = + li
dz dz dz*
dM_dty + 2;/ P<jJ = Q on z= _ h
dz 2 dz' 2 dz 3
= kJ kR/2/x on z — h
(12)
Hence we easily find
(13)
 . cosh kIi. t t> ■ i
4urf> =   . J,,kK sin i k?
rr *(sinh 2kA  2k/i) °
sink /c7i T r, i
— ; J „kK cosh kz
K(smh 2k1i + 2/c/i) °
, /, cosli k/i + 2k/; sinh «// T r> ■ i
^fl^ . . , , , —^r J„kR sinh kz
r K(smli2K/t2K/i) °
sinh kIi + 2k//. cosh k/i t „ ,
+ J,,kK cosh kz
K (sinh 2k/i + 2kIi) °
If these expressions, multiplied by e"* 5 , could be integrated with respect to k from
to oo , we should have at once a solution of the preliminary problem. But this
integration is not possible, owing to the nature of the functions of k near the lower
limit k = 0. In fact, if the values of 4.fi.ip, 4fi.6 in (13) be expanded in ascending powers
off, the expansions will contain terms in 1/k 3 and l/«r, so that near «• =
4fx<p = H/k 3 + K/k + terms of positive degree
4/x0 = L/k 3 + M/k +
These terms of negative degree are potentials contributing nothing to the stresses on
«=± h, as we see from (12), since kJ «:E contains no terms of negative degree. They
might therefore be subtracted from the expressions (13) without affecting the satisfac
tion of the conditions in (12). This simple subtraction would, however, introduce
terms not integrable right up to the upper limit, at least after e is put equal to zero,
as eventually it will be. The difficulty is met by subtracting from 4fi<p, not R/k 3 + K/k,
but ri> :i rKe K 7/c; and from 4^9, not L/k 3 + M/k, but L/^ + Me"*/*.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 145
(There are, of course, any number of equally suitable modifications ; instead of
e' Kh we might take e~ KC or l/(l+/c 2 ), for instance.) A solution of the preliminary
problem of normal traction equal to <?/(R 2 + e 2 ) 3 on the face z = h is thus obtained
in the form
4/*<£ =  / e ™\ . "^ K l J kR sinh kz
Jo ( K(sinh 2k/i — 2k1i)
sinh ah T n i H Ke _,cft  ,
+ / • i « 7 =rrT J oK K cosh kz  — > a.K
k (sinh 2/Ji + 2/c/i) k 3 k J
f cosh /c7i + 2/Ji sinh k7i,
\ k (sinh 2kJi — 2/Jt)
, sinh kIi + IkIi cosh k7i t t, i L Me~ Kh ) . ,,,,
+ — /.,«,■, — s^rr J o K K cosh kz — > cIk • • (14)
k (sinh 2»c/t + 2k7i) k 3 « J v ;
The solution of the original general problem is found by multiplying by
f(x', y') 1 2ir, integrating with respect to x', y' over the area A, and finally taking
the limit for e = 0. But a glance at the forms near k=co of the functions in (14)
shows that the triple integrals are absolutely convergent, it being supposed that
— h<Cz<.h. Hence we may integrate with respect to x', y' first, and by a well
known theorem the limits for e = may then be found by simply putting e = in
the integrands, provided the resulting integrals are convergent, as they manifestly
are.
This gives the value of (p, for example, in the form
/„ dK j fA z '> y'M*Wdy',
but, always provided h<.z<ih, we may change this if we please into
fff(x',y')dx'dy'rj(K)dK.
Finally, we may with great advantage confine our study in the first place to
what is usually spoken of as a unit element of normal traction at (x', y', h). The
area A enclosing this point is diminished, and the intensity of traction increased
without limit, so that j ff(x', y')dx'dy' remains equal to unity. The resulting
solution is simply that of (14), but with e put equal to zero within the integral
signs.
As we have just seen, the solution for the general case can at any time be
found from this elementary solution (15) by multiplying by f(x', y')/27r and integrating
over the area A.
4. Flexural and extensional components of the strain. Disadvantages of the
solution in definite integrals.
In the elementary solution each of the potentials (p, 6 may with advantage be
decomposed into an odd and an even part in z. Thus, for an element of normal
TRANS. ROY. SOC. EDIN, VOL. XLI. PART I. (NO. 8). 24
(16)
(17)
146 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
traction of 8^^ units at (x', y' , h), a solution is given by
<j> = <j) + </>,, ; 8 = 6„ + 8 e
where
, /"" j coah kJi _ _ . . 3z e" fe /2z 3 3R 2 z
y o I k (sinli 2*/;.  2k/&) ° 4x i 7t 3
» r* i cosh kJi + 2 kJi sinli kJi T t> • i 3z e~'
6 « = i — / • uo i — nr j ° kR sinh * z ~ T373 ~ —
.' i) I k (smh 2k/i  2k/;) 4^/^ k
A = I \  _ Jl"—— J kR cosh kz + — — [ cLk
^ Jo 1 K(sinh2K/; + 2K/;) ° 4k j
/, /"" I sinh k/; + 2k/i cosh k/( T r, , Se" 71 ) 7
6 = 1 { J ( ,KKcoshKZ > fiK
)» I K(sinh 2k/i + 2k/i) 4k I
The conditions satisfied at the faces by the partial solutions (16) and (17) are
easily made out. For when <f>, 9 are both odd functions of z, then «e, zy are even
and zz odd ; but when <j>, 6 are even functions of z, then z~x, 7y are odd, 7z even, as
is obvious from (5). Hence (16) gives equal values of opposite sign for 7 at corre
sponding points on z= ± h ; (17) gives equal values of the same sign.
It follows that (16) is the solution for elements of normal force of 4717* units at each
of the points (x\ y', li), (x', ?/, — h), the force being in the positive direction of Oz in
each case, and therefore a traction on z = h, but a pressure on z= — h; in (17) the only
difference is that the force is a traction on both planes.
Hence, also, (16) subtracted from (17) will give the solution for traction on z= —h
alone.
Each of the integrals in (16), (17) defines a potential function without singularity
at a finite distance in the space between the planes z = ~b %, and all the successive deriva
tives with respect to x, y, z of any of these functions may be calculated by differentia
tion within the sign of integration, provided we are dealing with a point actually within
the solid, so that — h<z<Ch.
The solutions defined by these integrals are therefore formally satisfactory. It is,
however, a serious objection to them that they do not lend themselves readily to inter
pretation, and it is not easy to make out from them any of the simple laws which the
ordinary approximate theory leads us to anticipate.
In particular, the solutions in their present form throw no light on the question of
the behaviour of the functions and their derivatives at points the distance of which from
the sources of strain is great in comparison with the thickness of the plate, a question of
great importance for the application to the thinplate theory.
The analytical transformations to which we now proceed reduce the solutions to a
form entirely free from these objections. Each of the integrals is shown to be composed
of two parts of very different character. The first part represents a function the value
of which diminishes with great rapidity as the distance from the source increases, while
the remaining part is a function of very simple form. Each solution is thus resolved
into a permanent or persistent element and a local, transitory, or decaying element, the
latter being insignificant beyond the immediate vicinity of the source.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 14f
5. Transformation of the definite integrals into series by means of Cauchys
Theorem.
The integral (p of (16) can be written as the sum of three integrals, namely —
[" p j cosh kIi sinh kz 3z 1/z 3 9 z\ I ,
*' = J o JoK 1 K(sinh2^2KA) IkW K \m TQhJf
)£(J>Be*)
(18)^
_f_ _9 z\r,T t, . „^^
8/i 3 40 fcy
It should be observed that the first and third of these integrals cease to converge when
K = 0. Hence the transformation does not apply to points on the line x = x', y = y', the
normal to the plate through the sources.
Consider now the first integral in (18). The function of k multiplying J kR within
the integral sign is an odd function of k vanishing for k = 0. Hence, as in (i), the
integral is equivalent to the complex integral
f ' r t? / cos ^ ^' s ^ n ^ KZ 3z 1/ z 3 9 z\ \ ,
J S ° I k (sinh 2Kh2Kh) + 4^/T 3 + AM 3 40 h) J '
the path of integration running from west to east along the whole of the real axis,
and just avoiding the origin, which is a singular point of G kR, on the north or upper
side.
On this path take points E, W at distances mr/2h to the right and left of the
origin, and on E W as side describe a square E W A B in the upper part of the plane.
The integral over each of the sides WA, AB, BE is easily proved to have zero for
limit when n tends to infinity through positive integral values.
Hence, by Cauchy's fundamental theorem, the integral over the path W E is equal
to the sum of the residues of the integrand at its poles in the upper half of the k plane,
multiplied by 2iri, that is, to the series
Zn r>/ \ cosh kIi sinh kz
G o* R (  ) T7 — uin T\ '
k *A(cosh 2k/i  1)
the summation extending over the zeroes of the function sinh 2*h — 2*h in the upper
half of the k plane, in the order of their moduli.
If £ n is a zero of the function sinh £— £, the corresponding zero of sinh 2K.h — 2«h is
K n = % n /2h, and
G /c, l R = G ^ = /s y^— e i e 2h approximately,
and we see that this part of (p , with its successive derivatives, is practically insensible
when R is a very moderate multiple of 2h. (Cf. § 7, infra.)
148
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
As to the other two integrals which occur in (18), we have proved in (h) that
These functions will occur so often that it will be convenient to reserve an invariable
symbol for the former of them, say
X (R), or simply x , = R 2 log —  ^R 2
and then
V 2 X = lo 8;
R
2h
(19)
The persistent part of <p is therefore
3z
Z 3 . 9 z
4A 3 * + W + 40 h) V ~ X '
which is the sum of two potential functions,
6. Types of the particular solutions composing the general solution.
A glance at the relation between the results just obtained and the form of <£ in
(16) enables us to write down at once the corresponding transformations of 6 , (p e , 6 e .
Collecting the results, we find
, y n p/ > cosh kIi sinh kz 3 / 1, , \ 9 «,
^ = zL G ,R(  ) Kh(cQah2Kh _ i ) ~ m (*X ~ 6 *V X ) + m ^x
6 n = ZG KR^ cosh Kh ' + 2 f sinh " 7 ') sinh KZ + ±(z x  l 2 3 v2 _ 2/fev 2 ) _ 9 2
■ ° K7i(cosh 2k1i  1) 4/A x 6 V X V X J 40/, V X
where k is a zero of sinh 2kIi — 2/ch, with positive imaginary part.
(20)
*.2g *B()?
sinh kIi cosh kz
+ lV 2 X
kJi (cosh 2 K h + I) 4
n _ p T?(sinh kJi + 2kJi cosh kIi) cosh kz _ 3 .,
~ « ° K k)i (cosh. 2 K h + I) ~i Vx
(21)
where k is a zero of sinh 2kJi + 2k1i, with positive imaginary part.
The solution must give w, zy, 7z all equal to zero at the two plane faces of the
plate, except when R = 0, and we are thus prepared to find that the strain defined by
the terms corresponding to any one root k gives zero stress across z = =fc h. Thus in
(20) 2 contains a series of particular solutions of the type
(i)
4>=  cosh k?i sinh kz ~F(x, y) i where (y 2 + k 2 )F = \
6 = (cosh kIi + 2kJi sinh k]i) sinh kz F(x, y) ) sinh 2kU  2k1i = J
(22)
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
'Calculating the stresses by means of (5), we find
 — + — ® 4 2z — f = (2k 2 7i sinh kIi cosh kz — 2k 2 z cosh nfi sinh kz)F
dz dz dz 2
—  — * + 2z — f = 2k' 2 (cosh k/i + kIi sinh «7i) sinh kz~E  2k 3 z cosh k/i cosh kz~F
dz 2 dz 2 dz 3 v '
both of which vanish when z= ± h.
We have further in (20) a solution of the type
149
(ii) *= 2 F + ^VF
From this, by (4), (5)
= ^f4*VF2^v 2 f K : ' ■" ,h '""' °
W =
d (
dx
w = —
w = (a+l)(Fi2VF) + 2(^ 2 A 2 )v 2 F
9=4 / ,( 2 2 /^V 2 F
(23)
The solution for unit normal traction on z = A contains a strain of this type, with
F = X (R) • 3/32ttM 3 .
Lastly, in (20) there is a solution of the type
(iii)
fj> = — zF ) Fa function of x, y
zY)
=  (a + 1 ),v
with v 2 F = ,
d¥\
dx
w = (a+l)F ;
Obviously this is merely a degenerate case of (ii).
Again in (21) we have a series of solutions of the type
(24)
(iv)
4> =  sinh K.h cosh kz F(x, y) ) where (v 2 + k 2 )F = )
6 = (sinh Kh + 2k/i cosh */<) cosh kz F(:c, y) / sinh 2/</* + 2k/< = f
(25)
In this, as in (i), z * = zy = zz = on 2 = ± h.
(v) The persistent part of (21) is of the type
giving
4> = F(z, ?/) ; =  3F(z, v/), where v 2 F = )
,eZF , .,^F ~ ~
" = {a ~ 3) ^' v = (a " 3) ^< ; ^ = 9 = s = )
(26)
150 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
7. Position of the zeroes of the functions sinh£=b£
The nature of the infinite series occurring in the above solution will be made more
intelligible by a short discussion of the position of the roots of the functions
sinh2x/i±2'c/i. These are obviously found from the corresponding roots of sinh £±£.
by dividing by 2h.
(i) sinh££=0.
£ = is a triple root, and the remaining roots are all complex, falling into sets of
4 of the form ±p±iq, where p, q are real. If, then, £=£ + *>?, £ and n being real, we
need only consider the case of £ and >; both positive. We have then
sinh $ cos 17 = £ and cosh £ sin 77 = 77 .
Cos n and sin 1 are therefore both positive, and n must lie between 2mr and 2nir 4 71/2
It is easy to prove that there is no root between and 71/2. For £ > tanh £, or
£/ sinh £> 1/ cosh £, so that
cos 77 > sin 17/17 or 17 > tan 17 or 17 > it/2 .
For every positive integral value of n, however, beginning with n—1, there is one
root, and one root only, with n between 2nrr and 2mtt + x/2. This will be readily seen
on roughly tracing the graphs sinh £ cos n = £ and cosh £ sin 17 = 17, or it may be proved
by an elementary application of the Theory of Functions. Thus, if we make the variable
£ describe the contour of the rectangle formed by the four lines
77 = 2w7r, f=N, i7 = 2ra7r + 7r/2, £=0,
where N is a large positive number, it will be found that the function v = sinh£ — £'
describes once a contour enclosing the point v = in the v plane. There is therefore
just one point within the rectangle at which v becomes zero.
For the large roots cos n must be small, or
77 = 2ra?r + tt/2  e, where e is small.
Hence
cosh ^ = r) — 2mr + tt/2 ,
or £ = log (4ra+ In) approximately.
Then
e = £/ sinh i=2 log (4?i + 1 ir)/(4re + 1 tt).
By successive approximation we may now find the roots as nearly as we wish, but
exact values are not at all necessary, the first approximation being quite sufficient for
our purpose,
£, = lQg(4n + lir) + C2n + i>rf .... (27)
(ii) sinh£+£=0.
In this case £ = is a simple root, and the rest of the roots are complex. If
%=$i + iy, we have
sinh £ cos 17 + £ = and cosh £ sin 17 + 17 = .
Cos»7, sin *j are both negative when £ and *? are positive ; hence >/ lies between (2n— 1)tt
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 151
and (2n 1)^ + ^/2, and it may be shown, as in the previous case, that there is actually
•one root with *i between these limits for all positive integral values of n.
Also
7] = (2n  1 )tt + 7J/2  e ; cosh £ = 2?nr  tt/2
e = 2 log (4re  1 t)/(4»  1 7r) .
To a first approximation
£„ = log(4ralTr) + (2ra£>ri .... (28)
In addition to the roots of (27), (28) we have of course a corresponding series in the
second quadrant, the images of these in the axis of imaginaries.
8. Approximate forms of the n th terms of the infinite series, when n is large.
It may be useful to give in terms of n approximate forms for the general terms of
.(20), (21) corresponding to the n th roots in the first quadrant,
(i) (p and .
kIi = \ log 4?«7T i (n + \)iri
sinh kJi = \e Kh = ■£(4?^7^)ie( n +}> , ™
cosh kIi. 1 4
e( M +lM( i)
«7i(cosh 2<c/i  1) sinh kU ■ 2 sinh 2 K/t (4?i;
z z
sinhxz = l(e Kh )h \(e Kh )~h
z z
( — TTlZ , TXZ
Hence in <p the general term
m(zh) *t* m(z+h)
4 i *—!l TTiizh) ZJ4 m(,z+h) )
= J G kR \ (4n*r) 2A e<»+i>— £—  (4«7r) 2A e(»+i)— IT" S ;
in O , the same as this, with the factor i/2mr omitted.
In both
G„»cR = f—e (»+Dx e 2A
(ii) (f) e and e .
kIi = £ log 4ra7r + (w  \)iri
cosh K h = i(4?jTr)ie( ,l i)' r,:
In <p e the general term
h^G " r 1 (4«7r) a*e(»J) \ +(4ra7r) 2A e("l)—ft— V
In O the same, with the factor i/2wr omitted.
, 2riR
9. The solution for arbitrary normal traction. Questions for discussion.
The solution of the general problem of given normal traction requires the multipli
cation of the functions in (20), (21) by f(x', y') and integration with respect to x', y'
over a finite area A. There is no difficulty in showing that these integrations can be
performed term by term, and that the resulting series converge absolutely.
152 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
When the solution of the general problem has thus been obtained in terms of series
of surface integrals, several questions present themselves for treatment, among which
may be specially mentioned
(i) For how many times in succession may these series be differentiated term by
term with respect to the coordinates x, y, z%
(ii) When the thickness of the plate is infinitesimal, but /(a?, y) does not vary as h
tends to zero, what are the orders of the various parts of the solution, and of the
related physical quantities ?
(iii) How are the answers to these two questions affected by discontinuity in the
applied traction, or its x, y derivatives 1
A perfectly general discussion of these questions would be tedious and difficult,
and it will probably be more useful to consider the points suggested in the light of a
special case, in which the integrations required can be performed, and the outstanding
features of the solution can be grasped with comparative ease.
10. Detailed solution of a special case. Term by term differentiations.
The solution we propose to work out is to satisfy the following boundary con
ditions : —
zz = 4:TTfiJ m (Pp) cos mm > on z = h
=  47ryw.J„,(/?p) cos mu) , on z =  h
— oii2=+/[, when p > a
zx = zy = , on z ± h .
> when p < a
p, w, z are the cylindrical coordinates of the point (x, y, z), so that x — pcoso),
y = p sin w.
/3 is any constant, and m is an integer.
The solution, obtained from (20) by integration, is
4> = <fy } + 4> 2 ; = 6 l + 6. 2 , where
*>= i( 2F l zVF  2 "^ !F )jo^ 2F • ■ <M>
with F = I /x(R)J„<(/V) cos mw'p'dp'ddi
, V / \ C0Sn kJ ' SiIin KZ f fn D T O ' ' 1 ' 7 '
92 = Zl \ ~ ;r; ^ — ;— ; rr I \jc,k\sA m ap cos m<a p dp day
a 'V (cosh k/i+ 2kJi sinli kIl) sinh kz , • , ,,
#2 = ^i , , .« , /, • (same mtesral)
(30>
the integrals being taken over the circle of radius a.
Consider in the first place the part of the solution defined by (p 2 , 6. 2 .
The value of the surface integral in (30) takes different forms when p > and <«,
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 153
As proved in (c)
9 9
when p<a, the integral = , i) u7r J m Bp cos mo> + „„ "* J m Kp cos mco(K«G m V<J„,/3tt  Q m KafiaJ „'(3a)
/J  K l ft'  K"
p>a, ,, = & m «p cos vio)( kuJ m r Ka J nl /3a  J m KaBaJ m '/3a) .
P 2  K Z
Now when up, kci are both large,
J mK aG m K P = — L_ e >;*(P«),
2 k ,Jap
which, with its derivatives, is very small when (p — a)j'2h is even moderately large.
Thus, in the space without the cylinder p = a, the part of the strain given by <p. 2 , 2 is,
when h is small, insensible except in the immediate neighbourhood of that cylinder.
The same remark applies to the strain within the cylinder, so far as it is given by the
parts of (p 2 , &> arising from the second term in the value of the surface integral.
We naturally inquire, how do these rapidly decaying parts of the solution behave,
and what is the order of magnitude of the corresponding displacements and stresses,
at points actually on the surface p = a1 Now, taking for example the value of (p 2
in the external region, namely,
<ft 9 = 2i (  ) f°. S 1 " \ S1 f ^ f x TZ— y &™ K P cos m<a ( KaJ m ' K aJ m /3a  J m Ka[3aJ m ' fta ) • • (30'}
k kIi (cos1i2k/i  1) p~ — K" \ /
we see from § 8 (i) that when p = a, the general term has the approximate form
A ( z Jlh. m(zh) z+h irtiz+h) )
A being independent of n. Moreover, each differentiation of (p 2 with respect to p
or z will remove a factor l/n from the general term. Hence three such differentiations,
but no more, are permissible, if— /^<z</i. But from (4) it is clear that none of
the displacements requires more than two, and none of the stresses more than three
of these differentiations for their calculation. As for 6 2 , the general term is of one
order higher in n than the corresponding term in <p 2 , but in compensation for this only
two differentiations are required to find the stresses. Hence, so far as the decaying
part of the solution is concerned, displacements and stresses at p = a may be found
by means of term by term differentiation, and subsequent substitution of a for p.
Again, considering the order of these various quantities in h, regarded as infinitesi
mal, and remembering that kJi and kz are of order zero in h, we see that the expression
for <p 2 in (30') and the corresponding expression for 9 2 are of order h 2 when p = a t
and each differentiation with respect to p or z diminishes the order by one. Hence
the displacements at p = a are of order h and the stresses of order zero, so far as they
arise from the decaying part of the solution (p 2 , 6 2 .
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 25
154 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
11. The same special problem. Summation of two infinite series.
An important part of the strain given by <p 2 , 6> 2 remains to be considered, namely,
that which arises from the term 27r/(/3 2 — k 2 ). J m ftp cos mw in the value of the surface
integral for the case when the point (p, w) is within the cylinder p = a. Denoting
these parts of <p. 2 , 0. 2 by <p. 4 , 3 , we have
<£ 3 = 2TrJ,„f3p cos mm £j( — )
cosh kIi sinh kz
(f3 2 K 2 ) K h (cosh 2k/41)
_ , ,, "V (cosh k/i + 2k/< sinh */t) sinh «,?
k (/}  k)k/i(cos1i 2/</i  1)
(31)
We note in the first place that <p 3 admits of three, and 3 of two term by term
differentiations with respect to z when — // < z < h, while x, y differentiations can be
performed without restriction. Combining this result with those already obtained,
we see that in the complete solution in terms of surface integrals, all the differentiations
necessary to give the displacements and strains or stresses at any point in the body
of the plate can be performed on the series term by term.
When h is small, <£ 3 and 3 are of order h 2 , and a z differentiation lowers the
order by one. This can be seen from the series, or otherwise, for, as we shall now
show, the value of the series can be found in finite terms.
Consider the function of k,
cosh kIl sinh kz
{j?  /c>(sinh 2kIi  2 K h)'
This function, multiplied by k, vanishes at infinity at all points of the path
E W A B described in § 5 ; hence the sum of its residues vanishes. The function being
odd in «:, the residues at the poles k = db k 1 are equal. Thus
2 (series of residues at zeroes of sinh 2kIi — 2kIi in upper part of plane)
+ 2 (residue at k = fi) + (residue at k = 0) = .
The residue at k = ft is
/ _ , cosh fih sinh fSz
'2/3(sinh 2/3h  2f3h) '
Also if
cosli k/i sinh kz A B , „
= = + (•••• , near k =
k (sinh 2kJi — 2kU) k 3
then the residue at k = is
Hence
2, i _ \ cos ' 1 "h s ' nn KZ _ cos h Ph sui h P z , A B
« ' (/J 2  K 2 ) K h (cosh 2<h  1) ~ y8 2 (sinh 2(3h  2/3h) /^ + /F'
It may be noted that A/ ft 4 + B/ft' 2 are simply the terms of negative degree in the
expansion of
cosh j3/t sinh f3z
P 2 {smh2ph2/3h)
in ascending powers of ft.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
155
Hence, putting in the values of A and B from (16), (18),
f _ cosh fflt sinh ftz 3z 1 / z 3 9 z \ \
<£ 3 = 2wJ fB #> cos mm <j ^(^2^2^) 4/3% 3 ^ \8P 40 ¥/ J
and similarly
„ T „ ( (cosh Bh + 2Bh sinh Bh) sinh Bz 3z 1 / z 3 , 69 z \ \
e 3 = 2,J m B P oo S m,^ BHs^2Bh2Bh) m^Ash^TOhJl
(32)
12. The same problem. Final form of the solution.
We come lastly to (p 1 , X of § 10.
The function F requires separate formulae for its expression in the three cases
m = 0, m = 1, ra > 1, but in all cases
F = (2Tr/B i )J m Bp cos mm + F 1 , when p< a )
F = F 2 , when p>a )
where F x and F 2 satisfy the equations v 4 5\ = , y 4 F 2 = 0. The values of F a , F 2 for the
various cases are given in (/).
When p<a , the term (27r/l3 i )J m 6p cos mw of F , taken by itself in (^ and l9 would
give
_3_ ^_JVz 3 1 Al
^ 4/i 3 /3 4 y8 2 \8)i 3 + 40 fc.
1 4/t 3 /3 4 + /3 2 W 3 + 40 ft,/ J
•2ir3 m Bp cos raw .
These are precisely the terms of negative degree (both in /3 and in h) with signy
changed, in the expressions for cp 3 and 6 3 given at the end of § 11.
If, then, we take this part of (p t , X along with cp 3 , 3 we have the complete solution
in the form
+ 2 7 r2()
cf> =  2tt J m Bp cos mm
cosh K.h sinh kz
cosh Bh sinh /3z
£ 2 (sinh 2/3ft  2/3ft)
08
^ ^— — — — — : —  J m «p cos mm(KaG m 'i<aJ m Ba  G mK aBaJ m ' Ba)
1  K z )Kh(cosh 2k/i1)
6 = 2irJ m Bp cos mw
(cosh /3ft + 2£ft sinh /3ft) sinh /Sz
and
/3 2 (sinh 2/?ft  2/?ft)
"V (cosh k1i + 2kIi sinh «ft) sinh kz t , ~ , T Q n T <a \
' 2 "f ( /j',.'H(cosh2,;'i) J "' cos '"™ ( "' G  "< ,J " /3 '' " ° M /3 " J " W
+ U z *> r VF   2,A ' !F  4^
when p<a .
(33>
* = 2ir2()
cosh «ft sinh kz
(/3 2  /c 2 ) K /i(cosh 2,<ft  1)
_8_ (
4ft 3
G m Kp cos mm(i<aJjKaJ m Ba — J m KaBa 3 m 'Ba)
*.>*».) + j^.
/i o "V (cosh k.1i + 2k1i sinh k/j.) sinh kz~, / t ' t o t „ o t 'o~\
^ = 2tt^ v — 77  — — — — —L — — — G mK p cos mm{KaJ m KaJ m Ba  J„,Ka BaJ m Ba)
K (B  K 2 )Kh(cosh 2k.1i 1)
+ 4JIK
«.j«^ p ."^0jjK^ p »
when p > a .
(34)
15C Mil JOHN 1KHJGALL ON AN ANALYTICAL THEORY OF
In (33) and (34) each line represents a potential function ; in (33) the first lines
define a particular solution giving the proper values of the tractions at the surface, as
may be seen from (13) ; the partial solutions given by the k series give zero surface
tractions, and represent a strain insensible except in the vicinity of p = a ; and the
solutions defined by the last lines, being of the form (23), give zero surface tractions.
From these remarks it follows immediately that the solution (33), (34) satisfies all
the conditions of the problem in the two regions p < a , p > a , taken separately. To
verify the solution completely, it would be necessary to show in addition that certain
conditions are fulfilled at the surface p = a, namely,
(i) that the displacements and strains are continuous at this surface, and
(ii) that the integral value of the stresses ex, zy ,%z over any small area lying partly
within and partly without the cylinder p = a, on either of the plane faces of the plate,
tends to zero when the area is indefinitely diminished.
The condition (i) ensures the ' synexis ' of the solution across the surface p = a, and
can be proved by showing, as may easily be done by means of summations similar to
those of § 11, that (p , 6 , =r and  T  are continuous at that surface. For by the Theory
dp dp
of the Potential this carries with it the continuity of all the derivatives of <p and , and
therefore of the displacements and stresses, as well as of all their derivatives, under the
proviso, of course, that — h < z < h.
The condition (ii), or some equivalent, is required in order to exclude the possibility
of stresses with finite resultant passing into the solid through the lines z = db h , p — a ;
or, in other words, in order to ensure that the solution is not partly due to linear
elements of traction at these lines.
13. Order of the various parts of the solution, ivhen h is small.
The final form of the solution, as exhibited in (33), (34) was obtained by combining
parts of <p x , #, with <p. 2 , 6 2 , and until this was done, it was not immediately evident
that (p and were potentials. Thus the part of the solution arising from the imaginary
values of k, or from any one of them, is not, within the region of applied traction, a
potential by itself, and the same is true of the (p^ , 1 part, which may be considered as
coming from the zero values of k. This has sometimes to be taken into account in
calculating the stresses ; the formula for zz, for example, in (5) requires additional terms
if, while u, v, w are still given by (4), (p and 6 are not potentials.
On the other hand, the separation of the solution into the two parts (29), (30) has
this very marked advantage that, when h is very small, the first part gives the terms of
the two lowest orders in h of <p , , namely those of orders h' 2 and h°, while the second
part, as we have already seen, contains no terms of lower order than h 2 . When, how
ever, we come to calculate displacements and stresses, the separation is less simple,
mainly in consequence of the fact that x , y differentiations do not change the order of
<p } , 1 , but diminish the order of <p 2 , Q., by one for each differentiation.
THE EQUILIBRIUM OK AN ISOTROPIC ELASTIC PLATE.
157
The following table, which may be deduced immediately from the results of
SS 10, 11, shows the order in h of displacements and stresses arising from 2 , $ x and
(p., , 2 respectively.
U , V
IV
xx , xy , yy
ZX, ZIJ
«£ 2
» #2
*!.*!
p2«
p = a
2,0
2
1
3, 1
1
1
2,
1
1
It thus appears that the first part of the solution gives all the displacements to a
second approximation, and all the stresses but zz to a first approximation. With regard
to T z , it should be observed that the solutions depending on F x , F 2 contribute nothing
to it, so that, within p — a, its value comes altogether from the particular solution,
and without p = a, its value is zero beyond the immediate vicinity of that surface.
14. Methods and results of the special case extended to the general problem of
arbitrary normal traction.
One feature of the solution expressed by equations (33), (34) we have already
found useful, especially in the important case when h is small, in such a way that fih
and h/a are small fractions. We refer to the explicit separation in the solution of a
purely local element, entirely negligible except within a certain strip of breadth com
parable with the thickness of the plate, from an element of a persistent or permanent
character, with an area of influence not affected by the indefinite diminution of h.
Another advantage of the form of solution in (33), (34) is that the particular
solution for the space within which the traction is applied is found in such a form that
it can be readily expanded in powers of h, so as to give the terms of positive order in
the infinitesimal h, as well as those of negative order which were already separated
in (29). Thus in the particular solution, or first line of (p in (33), the factor
cosh [ilu sinli f3z
/3 2 (sinh 2/3/;.  2 /3k)
can be expanded in ascending powers of (3, the series converging if j 28h  <  ^  , where
^ is the complex root of sinh £— £ = with smallest modulus. Since z is of the same
order as h, and we are supposing /3 independent of h, it is clear that the terms of the
series will be of ascending order in h.
We shall now show how the solution for the general case when the given normal
traction is a function of x , y of unspecified form may be transformed, under certain
restrictions, so as to yield the advantages to which we have been referring as pertaining
to the solution (33), (34).
The problem we suppose to be the same as that stated at the beginning of §10,
158 MR JOHN DOTJGALL ON AN ANALYTICAL THEORY OF
but with f(x , y) instead of J\„fip cos ma , and with any continuous area A instead of the
circle within p = a.
The solution will then be of the form defined in (29), (30), but the integral of (29)
will now be
f = ffx(W(*, y'Wdy'
and the integral of (30)
I = jJG (KR)f(x',y')dx'dy:
Since ^ 2 x (R) = log(R/2h) we have
V*F = V 2 (V 2 F) = 2*f(x,y)\
also (v 2 + « 2 )I= W(x,y) ] W
If f(x , y) and its derivatives of the first two orders are finite within A, we may
transform I by Green's Theorem. Thus, excluding from the area A an infinitesimal
d 2 d 2
circle about (x , y) as centre, and writing v 2 for ^ 2 + ^7 2 ,
I =  J jf(a:',y')v' 2G o KRdx ' dl J
or
1 =  i 2 f JG KRV%x',y')dx'dy'
 y { /(*'• ^°o«b  G o" R J/( a; '> y } ds \
 2 4f(*,y)
K
the line integral being taken round the boundary of A.
If this threetermed equivalent of the integral I be substituted in the series for
<f>2 and 2 , each of these series may be subdivided into three, (f> 2 for instance into
(ft T x cosh Kh sinh KZ [ (a k r WH(x' v'\dxdv
k 2 /c/t(cosh 2kJi  I) j
a series of the same general form as the original series, but at once more convergent,
and of two orders higher in h ;
z . . x "V 1 cosh kIi sinh kz ( f ., , i\ d n „ n y, d ., , , x I ,
( u ) ^ K fe(cosh2^i) J \f(*,y)^o«Kz^,f(*,y) \d»,
which corresponds to a strain local to the boundary of A ;
,... x , "V 1 cosh kIi sinh kz ■ , • , , , . .,
(in) 2irf(x, y)Zj ~2 T7 — ,i .; ; _ i \ > a senes which can be summed in the same way
as (31), being in fact simply the first series of (31) with /3 = 0. The sum is therefore
cosh k!l sinh kz
2irf(x , y) ■ coeHicient of k° in
K 2 (sinh 2xh  2k!i)
We may now, by repetition of the same transformation, obtain a similar threefold
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 159
equivalent for the series (i), and continue the process as far as the continuity of the
derivatives oif(x , y) will permit. We should thus obtain, in place of <p 2 ,
... , , V, ,1 cosh kJi sinh kz f f ,„ , , , ,
(l) (  )\f (  is K/f(cosh 2jj r i } j j G kR v "/(* , V )d* dy ,
a function of order h 2n+2 .
(ii) A series of lineintegrals which we need not write down, corresponding to a
local perturbation at the edge of A, and giving the edge values of the relative part of
<p up to terms in A 2 ".
(iii) A series of n terms
27r ( C o/ C 2V 2 /+ cMf + (  )" _1 C2n2V 2 "^y) ,
where c 2r is the coefficient of x ir in the ascending power expansion of ( ) 2/ • i 9 * _ o aV
and is obviously a rational integral function of z and h of degree 2r + 2. If the function
f(x , y) has its derivatives of every finite order continuous throughout the area A, the
process can be carried to as high a value of n as we please, and we can thus obtain the
values of (p 2 , $2 to an y required order in h. It should be noticed, however, that the
series (iii) is not necessarily convergent when continued to infinity, as we may see by
taking as an example f= cos ax, when the series would become 2w cos ax(c + c 2 a? + c 4 a 4 +•■•),
which is the expansion (without the terms of negative degree) of
n cosh ah sinh az
27rcos«u? — — ,
a^(sinh Aali — lah)
and is therefore divergent if  2a h \ >  ^  , ^ being the complex root of sinh £ — £ =
with smallest modulus. The form of the condition suggests that in ordinary cases the
series will be convergent if h is small enough ; and when this is so, this part of <p 2 , 2
taken along with <p x , X will define an exact particular solution within A, giving the
proper values of the surface tractions, and arranged in terms of ascending order in h.
As a special case, the series will terminate if. for some finite value of n, ^/ 2n f^ ,
and in particular if f be a rational integral function of x, y. (It may be noted here
that the solution for f= p m+ip cos moo might be obtained from the solution for
/= Jmftp cos mw by expanding in powers of /3, and equating coefficients of / 8 m+2 ^ in
conditions and solution.)
Looking back now to the ^ , 6 1 part of the solution, and having regard to (18),
(29), (35), we see that we may write symbolically
F= 2tt V  4 /; v 2 F = 27TV 2 /, and
^ =.2 fl (c_ 4 v 4 /c_ 2 v 2 /).
The particular solution to any order in h is then given by
cj> = 27r(c_ 4 vyc^v"y+c /f 2 V 2 /+ — )
where c 2r is the coefficient of K 2r in the expansion of (  ) . °° s , ^ s ™ Jf* , for negative as
well as positive values of r ; or, as we may put it, this particular part of (p is given by
expanding (  ty jj^^kh* 2Z1) » writin g ~ V 2 for K \ and operating on/(x , y).
160 MR JOHN DOUGALL ON AN ANALYTICAL THLOBY OF
15. Independent symbolical solution of the general problem.
The form of the last result suggests a method of dealing with the problem from
the beginning, which, though not easy to develop independently with thorough rigour,
has the advantage of conciseness, and will therefore be useful in giving a rapid account
both of the foregoing solution and of those to be obtained in the following pages.
We begin by observing that
(£ + iif + £) sinh *■* v) ' sinh k2 CS + $ + K *) f{x ' y)
= sinh kz(v 2 + K 2 )f(x , y).
Hence sinh <zf(x , y) is a potential function, provided we regard k as an operator such
that ie 2 = — v 2  We may, if we please, take k = i^j , but it will not be necessary to
interpret odd powers of the operator y.
On this understanding, it is obvious from (12), (13) that we obtain a solution
giving
ZX = ZiJ — 0112= +/l
^l = on z= — h
= j\x , y) on z = h , within the area A ,
by taking
cosh k!i sinh k/i , \
4 ^ =  K*(8inh2K/ 4 2icM Sinh ""'J " ^( s inh2KA+2 K /^ COSh "*/ \
. (37)
* 2 (sinh 2 K h  2 K h) V K 2(sinh 2xh + 2k//)
cosh kIi + 2kJi, sinh kIi sinh kIi + 2k1i cosh k!l
^ " ^( S inh2^.2^) sin W + J^iiDhSrt + Si*) C ° ShKZ y )
Now, taking as a specimen the first term of iiu<p , we observe that the function of k
cosh kIi
(  ) „, . , , — =t\ sinh *z
' K 2 (sinh 2k/i  2kA)
vanishes at infinity round the path W A BE of § 5.
Hence the function is represented by the sum of its polar elements. (If k 1 be
a simple pole of the function, and if in the vicinity of this pole the function
= A 1 /(k — k 1 ) + finite, then ^/(k — k^ is the polar element at this pole, and A x is the
residue there. The point k = is a multiple pole, and the polar element there has the
form A/k 4 + B/V 2 , these being the terms of negative degree in the expansion of the
function near k = 0).
Taking the elements belonging to zh Kj together we obtain
cosh kIi sinh kz
().
/c 2 (sinh 2k1i  2k1i.)
A B . / 1 1 \ ,
T + — + A, + •
K* K" \K — Ky K + Kj/
A B 2A.K,
K K K — K,
A B . cosh K,h sinh k,z
+ + ( ) l '
1
2 2
the series extending over the poles with positive imaginary part. When we put
k 2 = — y 2 , this part of 4/xc/> becomes
Ay 4 / Byy + 2j w — Tin — Tv "T o • f,
x J v • K k/j(cos1i 2k1i I) y  + K 2 •/
where k is no longer an operator, but simply a root of sinh 2kIi — 2kIi = 0.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
161
Now since
one value of
( V' 2 + k 2 ) J JG kR/(z', y')dx'dy' =  2tt/(x , y) ,
V + k ZirJ J
Similarly one value of v  !/' is — j J\og(R/2h)f(x, y')dxdy (38)
and of v*/ ~f i (  K 2 log 2*  I R2)/(x', z/)^W
Hence for the first part of 4p.<p we obtain
+ 2^ <  ) K/,(cosh 2k/, 1)]] G o«W(x , V )dx dy )
which agrees with our previous solution, the element of which is given in (20).
Further, the results obtained at the end of § 14 clearly agree with what we should
get by expanding the function of k in (37) in ascending power series and interpreting.
As an example of this use of equations (37) we may find to a first approximation
the value of zz at points not very close to the edge of A.
From (5)
(39)
m = MW)it**) + *&*+
i 2 /. „\ d 2 f A ,\ . d s t
2Z dJ
(2 cosh kJi + 2k/(. sinh «//) sinh kz — 2kz cosh kJi cosh kz
sinh 2k1i  2kIi
(2 sinh k/i + 2kIi cosh k/i) cosh kz  2kz sinh k/i sinh kz
f
+
sinh 2kJi + 2 k/i,
Thus
= (3h 2 z 2 )zf/2h*+/.
? z ={(3h 2 z 2 )z/lh*+l/2}f(x,y),
and we verify at a glance that this gives the proper values at the faces.
(40>
16. The problem of tangential faoe traction. Solution for an element of traction.
We will now pass to the problem in which the given surface traction is tangential.
Taking the direction of the traction parallel to the axis of x, we may take for
conditions
**■ = /0 c ,y) on z = + h 
on z =  h >
on z= ±h )
(41)
zy = zz =
According to the method explained in § 3, we begin with the function kJ (kR) in place
of f(x , y), and determine potentials \k, 6, <fi giving
ar$ d 2 6 d 2 <j> d 3 <f> 1 _ _
r r + 5 — r +7j+ 2z , ., ■ = 7 r K J n KR on z = h
dy dz dxdz dxdz asrdx 2/x °
_ dM_ d°6 dcf> d 3 <f>
dxdz dydx dydz " dzkly ~
dz 2 dz 2 " dz 3
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8).
on z= —h
on z= +h
on z= ±li
(12>
26
162
MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF
Since J,.kR = — . ., J,.kR — , t^J„«R, it is clear that these equations will all be
k (/.<• ° k~ ay ° ' 1
satisfied if we take
where
d\l/' a >W dcf>'
d *' l 1 T P 7 )
— t =  s J n*K on z = li \
dz 2fj. k ° r
= on z = — h j
+ ; + 2?: ,t = K J„«R on z = A
dz dz
dz 2
d 2 ff d^' . d 3 <f>'
— —  4 2z — — =
dz 2 dz' 2 dz s
on z s= — h ,.
 +h
on
(43)
(44)
(45)
From (44),
4 ^'= ? s°inh2L J o kE '.
or, separating the odd and even parts in 2,
, 1 cosli k2 1 sinh kz
T k^ smh kIi ° k* cosh Kft u
We also find easily
Att<$> =
sinh kIi
J„kR sinh kz
K 2 (sinh2/c/; 2k7i) u
cosh kIi
97 — , o 3 , , J n KR cosh kz
K 2 (sinh2K/i+ Zk/i) °
(46)
2«7i cosh k/;  sinh kIi .
4u,P = 9 / • 1 n 7 n 7 \ J,,kR Sinh K2
^ /< (smh 2k/;  2k/<.) °
2k1i sinh k/7  cosli k]i
<LkR cosh /c2
K(smh 2/c/i + 2k/i) °
Treating these expressions as in § 3, we find a solution for an element of Xtraction
at (x', y', h) of Stt/ul units given by (43) with
3inh kz
f:
k 2 cosh /</(' "
cosh kz
J„kR + e* h  \dK
1 e* h / 1
r ( cosh kz T ^ 1 e" fl / 1 1 1 \ )
J „ ( k sinh km ° ni k/i V 2 6 4 //
r" f • sinh kA sinh kz 3z e  *' 8 / 3 1 \)
* = j i ~ K^mT2^2WTy'» KR + « + ^< " ~2 Wz ~ J hz ) J dli
r ( cosh kK cosh kz 1 e K ''/ 1 1 \ )
/ „ j K(smh 2ic/i + 2k//) ° i^/t. &k/i \ 2 3 / j
['■ ( (2i<h cosh k//  sinh k/i) sinh «.; 3z e K, V 3 19 x i
° j n \ K'(s>nli 2kA; 2,<h) J " kR  4^P ■" 8*W  2 R " + h h Z ) ) dK
K 2 (*n\\i2Kh2Kh) u o R " 4k% 2 8/c/i'V 2"'^^
r r (2«A sinh k//  cosh k//)cos1i kz t ^ 1 e~ Kh / „ 1 „ 11, A)
+ j„ 1 ^(sinh2^ + 2 i ) J «" R + 4^ + 8^( 22  2 E2 ~ T 7i2 j } ^
(47)
These expressions may be transformed by the method of § 5, and a slight inspection
of the relations between (16), (17) and (20), (21) will enable us to write down the
(49)
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 163
results at once. These are, if we separate the parts odd and even in z,
*»' = ?< 2 )?S^S »(« E ) + ^x} .... («)
where k is a positive imaginary root of cosh ah,
y sinh *7i, sinh kz 3 / 1 \ 1
^o = 4 j(_) ^(cosh2K/ i l) G « KR  M" X " 6* V 2 XJ40 Z V 2 X
Z(2kA cosh k7i sinh /c7;)sinh kz . _, 3/ 1 „■,«, n \ 1 n
^( g osh2kfel)  G o( kR ) + «<*  6*Vx27^v 2 x) + 4o*VX + zV 2 X
where k is a zero of sinh 2di  2<h , with positive imaginary part.
where k is a positive imaginary root of sinh kK.
y cosh *cA cosh «z 1 / 1 \ h
*' = 4*< " >«%(coah 2«* + 1) G ° kR ~ 47il X ~ 2 z " v *,) + 24™
•^ (2/c7i sinh k7i cosh k7?) cosh kz 1 / 1 \ 117i
'• = 4 K%(cosh2KA + l) G o(^)^X  2 z2 ^ 2 xJ24 V 2 X
where k is a zero of sinh 2<h + 2k]i , with positive imaginary part.
The solution is defined by these equations with
+ = *£ e = ^,4>=^L.
dy dx dx
17. Composition of the solution.
On examining the composition of the solution, we observe in the decaying parts of
(p , , solutions of the class already obtained in (22), (25), and in the corresponding part
of \^ , solutions of the type
•A = sin(2» + l)gF(a;, y)
(51)
^ = cos— F(x, y) ,
d\p _.
each giving % = on z = + 7i , and therefore zero tractions at the surface. As for the
permanent terms, they may be arranged in the following groups, in each of which the
surface stresses vanish.
zi 1 c7 9
1(54
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
These are of the types (23), 24).
(iii)
d 
dy
6  ^ v "*
This gives, by (4), since y 4 x = 0,
( P
W =  *dxdy V X
d ,
and may therefore be considered as of the type (24). It is important to note, how
ever, both here and in the cases immediately following, that the transformation on the
value of u will not hold after the elementary solution has been integrated for the pur
poses of the general problem in (41).
(iv).
* h dy\ K
1 J»„S
T ^V~X
Id/ 1 ., ,
U dx\ x ~ Y zvx
ill dxV
(,
¥^
Now F being any function of x, y, satisfying y 4 F = 0, the solution
leads to
d?F
dy 2
d*¥ .3  a „ d 2
+ <" +1 )^ +i i v iv^
?o
and z./ = ~y = zz =
(y)
<A
dy
h d
dx dy
h d „
which may be further decomposed into
U dx* x >
Uh d_
24 dx
V 2 X
of the form (26), and
the displacements corresponding to which vanish.
h_ d 2
(52)
(53)
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 165
18. General solution. Comparison with the solution for normal traction.
The solution of the general problem of (41) may be found by multiplying the
expressions for >//, f , </>' in (48) .... (51) by —f(af,y') and integrating term by term
over the area A within which f is finite. As in the case of the problem of given
normal traction, term by term differentiations of the resulting series will be legitimate
just so far as the derivatives are required for the calculation of displacements and
stresses. In order to see this, it should be noticed that, while an extra differentiation
as to x or y will be required in virtue of (43), the series for \f/, &, cj>' have general
terms of one order higher in 1/k as compared with those of (20), (21). One effect,
however, of this additional differentiation will be to increase the relative importance at
the edge of the area A of that part of the displacement and stress which arises from
the local perturbation, such displacement being of order h, and stress of order zero, as
in the former problem, whereas the displacement and stress as a whole is of higher
order in h than before.
The functions \/, 6', (p' being symmetrical about the axis R = 0, it is clear that the
solution for an element of traction of 8717/ units at (x', y f , h) parallel to the axis of y is
given by
* ft', 6 = f,<j, = d jL (54)
ax ay ay
withx//, 6', 4/ as in (48) . . . . (51).
It will be seen presently that surface traction may be regarded as a special case of
force applied in the body of the plate. We may therefore postpone any more extended
development of the above solution, and in particular any more explicit comparison of
the results with those of the accepted approximate theory of thin plates, until we
have obtained the solutions of the problems relative to sources of strain situated in
the interior of the solid.
19. Normal force applied at a single internal point. Solution
in definite integrals.
We take first the case of a single force, say for convenience of 4^^(0 + 1) units,
applied at (x 1 , y', z') parallel to Oz , the faces of the plate being free from stress.
Referring to (6), we see that the conditions of the problem may be taken to be
(i) « = ( 2 '^
(IX
v = (z'rzfl +V . (55)
dy
(ii) U, V, W, along with their derivatives as to (x , y , z) of the first order, are finite
and continuous at every point of the solid at a finite distance, and have derivatives of
+ u
+v
+ ar~
1 + w ;
166 MR JOHN OOUGALL ON AN ANALYTICAL THEORY OF
the second order satisfying
/*V 2 U + (A + p) d * = ; ^v 2 V + (A + /x)^ = ; /, V 2 W + (A + ^ = j
(fx ay dz
where
. _ dV dV rfW
rZx rty dz
(iii) zx = zy = 7z = on « = ± A,
It is clear that these conditions do not completely define the solution, seeing that
no condition to be satisfied at infinity is mentioned. But instead of laying down any
such condition at infinity, it is simpler in the first instance to be content with any
solution fulfilling (i), (ii), and (iii). The most general solution can then be obtained
without difficulty, and with this before us, conditions at infinity can be discussed to
much greater advantage than at present.
The problem is solved when U, Y, W are found in the form (4), so as to give
the same tractions on z=dzA as those due to (6), but reversed. These reversed
tractions, as follows very readily from (5), (8), are given by
1  a i , / '\ dr' 1
^ T +{Z  Z) Hz
zx _
_ d
%J~
dx
5/_
. d
2/x
dy
Now when z > z',
1 + adr 1 . _ kcPti
2/jL 2 dz ^ Z) dz 2
e^^JoKRd/c
J
dr
but when z < z',
J C
2,.l f
dz J o
dz*
^^JokIWk
7V
dz J o
( ^Z1 1 = r K *e«*^J KRdK ..... (56)
dz' J o
Hence if U, V, W be defined as in (4), the function \/ is not required, and the
conditions to be satisfied by 6 , cf> are, if in the first instance we take integrands
instead of integrals,
dz dz dz ! t 2 ' )
= i l ~   K (h + z) i e^+ z ')J KR, on «=  h
= {  Lt?K  K%h + z') i e «(*+^J kR, onz=/t • . (57)
( <4 )
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 167
Assuming
(j> = A sinh kz + B cosh kz 
6 = C sinh kz + D cosh kz / °
J/,kR
(57) give four equations to determine A, B, C, D. By addition and subtraction
these are resolved into two equations for A, C, and other two for B, D. Thus we
find
, sinh kzJ„kR f  • i > , i / , 9 i « , \ )
<t> = —7 . . „ , " . , { kz sinh kz  if cosh kz (e' iKn + a + 2k//,) }
K(sinh 2kJi  2k//,) ( )
c osh kzJ kR ( kz , cQgh ^, , ginh KZ '/ e _ 2Kfi _ a _ 2k/ . I
K (sinh 2k//, + 2k//) I ; \ j
= f" hKZ «V K J _ KZ ' s i n h K g'( e  2«/> + 2/c70 + cosh kz{— e ' *«* + — + a/c^ + 2k% 2 ^ I
K(sinh 2k//  2k//,) I \ 2 2 J )
cosh kzJ,,kR f , h kz ,, _ 2Kh _ 2k] , ginh J _ a_  2Kh + }_ + j + 2k2 ^ \ \
K(sinh2K/i + 2K//) I v ' V 2 2 J)
. (58)
If these expressions could be integrated with respect to * from to oo , the balancing
displacements U, V, W of (55) would be determined. Near the upper limit the
functions converge to zero exponentialwise, since both z and z' lie between — h and + h.
But for k = both functions are infinite, and their expansions in ascending powers
of « contain terms of negative degree which must be removed after the manner of S 3.
The integrals are then convergent, but a further modification of a different sort is
1 necessary before they can be transformed into series as in §§ 5, 16. The possibility
of this transformation in the former cases was intimately related to the fact that the
functions in (13), (46) were odd in k, which the functions in (58) obviously are not.
However, when the odd and even parts are separated, the latter are found to have
a very simple form, free from the denominators sinh 2k1i dz 2*A, for we find
<j> = —sinh k{z  ;/) J kR
2k
sinh /czJ kK ( / • i / i / i c 7 x i  )
+ . . . , , ' ,. < kz sinh kz  A (a + cosh 2k//) cosh kz >
K(smh 2k1i  2k//) I z x ' J
, COsIikZcLkR f, ,/i/ in, xi I
+ , . , n = — — ; . < kz cosh kz + A (cosh 2k//  a) smh kz >
K(smh 2 k//, + 2 k//) I i v ' J
=  \ ~ sinh k(z  z) + z cosh k(z  z) \ J kR
+ , ■ ■, r.~T " ~ , l ~ kz' cosh 2/c/t sinh kz + ( — cosh 2k// + — + 2k"//, 2 )cosh kz >
K(smh 2k//,  2k1i) I \ 2 2 / J
+ / • i — > \ " K n , . i kz' cosh 2/c/i cosh kz' + (  — cosh 2k// + —  + 2k'//' 2 )cosh kz' >
K(smh 2 k//. + 2 k//,) I V 2 2 / j
•  • (59)
The even terms in «■ can be eliminated from these expressions by including the
168 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
values of (p , 6 which d« fine the source, given in (8). For these are
<£= i. j'  e ^Oj oK R _ e " h \—, if z>z'
=  if"  e *(z')J oK R  e^ 1 — , if .:<./.
61 =  — I { e " « 2 " *") J kK  e " «* i !  + ::' j e " "< z " *>J kR^, if 2 > 2'
2 .' ip ( ) K JO
= \\ i r*r.z),) oK \i _ eKft I ' 7/< + ./ I %^OJ oK RcZk, if e<z'.
(60)
These will obviously be reproduced, after preparation by removal of terms of negative
degree and integration, from
«■/,= —e«( z  z )J KU, if z>z
2k
=  J_e«( z  2 ')J KR, if z<z'.
2/c
_5L + ' y^OJ oK K, if. j>;.'
A +z ' W*')J>R, if z<z'.
When these last terms are taken in, the first lines of <p, in (59) become
<t>= ± — cosh k(z— z')J kR
'2 k
6 = + ■£ cosh k(z  z') J kR + z sinh k (z  z') J kR ,
the upper or lower sign being taken in the ambiguities according as z > or < z'.
Hence, when the source is taken in, the following are the unprepared and unintegrated
forms of (p , : —
<j> = ± —  cosh k(:  z')J kR
2k
sinh kz.J„kR f , . , , ,, 1 o ; \ i > )
+ — — ■ ; < kz sinh k.v  Ma + cosh 2k/i,) cosh kz >
k(siii1i 2k/j  2#cft) I J
+ —.,  , uK . < kz' cosh kz + i(cosh 2k1i  a) sinh kz' \
k(siiui 2k1i + 2/cA) I J
6 = + — cosh k(z — z'). I kR + z' sinh k(z  z')J kR
2k
+ / • t « t ° ft , v ^  K2 ' cosn 2k ^ sinh kz' + ( ^ cosh 2k/j +  + 2k 2 /< 2 jcosh kz }
K(smh 2k/i  2/c/i) ( \ 2 2 / J
+ .°. S , jf ,°, a , . \ kz cosh 2kU cosh kz' + (  ^cosh 2kJi +  + 2k 2 A 2 \wh. kz \
k(sdi1i 2kIi + 2kIi) t \ 2 2 / J
(61)
In (61) the terms of negative degree in k are of the forms: — in (p , H//c 3 + K/fc;
in 6, L/k 3 + M/V ; and these terms, as in § 3, give zx = zv/ = zz = at the faces of the
plate.
(62)
THE EQUILIBRIUM OP AN ISOTROPIC ELASTIC PLATE. 169
Hence if from <p in (61) we subtract H/^ + Ke*"/* , and from 6, L/^ + Me"^, the
resulting expressions, integrated with respect to k from to oo , will define a solution
of the problem stated at the beginning of this article.
20. Normal force applied at a single internal point. Solution in series.
To the integrals thus obtained we can apply the transformation of § 5, but one
remark should be made. From the synthesis which gave (61), it is sufficiently obvious,
in view of the forms in (58) and (60), that the expressions of (61), with Gt kR sub
stituted for J (kR), vanish effectively at infinity in the first quadrant of the k plane ;
that they similarly vanish in the second quadrant follows at once from the fact that the
functions of (61) are odd functions of k.
A glance at the relation between (16), (17) and (20), (21) will again save us the
necessity of writing down the details. Thus, let the values of H , K , L , M , when R is
put equal to zero, be denoted by H , K , L , M . Then the persistent part of the
transformed solution is given by
4> = H oX (R)  K oV 2 x(E) I
6 = L oX (R)  M oV 2 x(R) ] '
The decaying part is given by
d, = 2^,,® — , „ ., — ,\ \ kz sinh kz  7r(a + cosh 2/</t)cosh k?J \
^ K K/i(cosli 2k/i  1 ) [ 2 J
= y\ 7 T ,° K , S "Y K '\ \  kz cosh 2k1i sinh kz + ( ^ cosk 2/c/i + ^ + 2/< 2 /t 2 jcosh kz } > . (63)
, k/i(cos1i 'lull  1) I \2 A J ) ;
= y, ~rr ,  , — Vs \ kz sinh kz   (a + cosh 2k/i)cos1i kz I (  cosh 2kIi)
* Kh (cosh 2k/i  1 ) ( 2 X J v '
where k is a zero of sinh 2/c/a — 2kJi , with positive imaginary part ;
with
<A = 2li t /°" i o t K \\ \ kz cosh kz + „ (cosh 2k/i  a)sinh kz \
^ K K«.(cosh 2k/i+ 1) ( 2 X )
= .Zj , ,° K . n , K ~ { kz cosh kz + ^ (cosh 2k1i  a)sinh kz \ cosh 2/Ji
k /c/i(cosh 2kIi+ 1) I 2 X J
where k is a zero of sinh 2kJi + 2k1i , with positive imaginary part.
When the values of H , K , L , M are obtained from (61), it will be found that
(62) may be decomposed as follows : —
(ii) % I  %\ } each multiplied by J^  tf ) + 3 i^( J W  J *)
(iii) 4> = "^v 2 x = ^
(iv) <£ = + iv 2 x> 0=±£<*v 2 x> with upper or lower signs, as z is greater
or less than z'.
(65)
TEANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 27
(64)
170 MK JOHN DOUOALL ON AN ANALYTICAL THEORY OF
There being no discontinuity of displacement or strain at the plane 2 = 2', except at
the point where the force is applied, we are prepared to find that (iv) give no displace
ment at all if R> 0. So long, however, as we keep to the specification of the strain
by the <p , 6 functions, it is convenient to retain the terms in (iv). By so doing, we of
course make the <p of the space above the plane z = z' and the (p of the space below that
plane two distinct potential functions, but we preserve the nonsingular character of
each of these functions at the axis E = .
If we take the limit of the above solution for z' = h, which obviously may be done
by putting z' = h in each term and using the lower signs in (iv), we obtain simply the
solution of (20), (21) multiplied by ^(a+1). Since the present solution is for a force
of 47r / u(a + 1) units, and the other for an element of traction of 8^^ units, it follows that
a unit element of traction may be regarded as simply the limiting case of a unit force,
the point of application of which approaches indefinitely near the surface.
21. Solution of a special problem of internal areal normal force.
When the displacements due to a unit Z force at (as', y', z'), with the surface free,
are known, the corresponding displacements for a body distribution of force, of
amount Z(x', y', z') per unit volume at (x f , y', z f ), can be found by multiplying by
Z(x r , y', z')dx' d'f dz' and integrating through the space in which Z is finite. Certain
peculiarities in the form of the solution given in ^ 20 make it convenient to take the
integration with respect to z' last, or, as comes to the same thing, to begin by con
sidering the solution for an areal distribution of force on the plane z = z 1 ', of magnitude
Z(x', y r , z') per unit area.
We take first a special problem analogous to that worked out in § 10, and suppose
the Z force to be distributed over the area of a circle of radius a in the plane z = z\ with
centre on O2 , the intensity per unit area being 47^(0 + l)J m (3p cos moo. It will be
sufficient to attend to the value of <p, for when that is known, the corresponding value
of 6 can be written down at once.
The series deduced by integration from (63), (64), say <p 2 , fall naturally into two
parts as in § 10, viz., (i) series defining a local perturbation at the cylinder p = a,
4, = ^ Kftfcosh 2k}i 1) \ KZ ' Sillh KZ ~ Ty ( a + cosh 2k7(.)cos1i kz'  . ^" k2 P« (where sinh 2kU  2kJi = 0)
+ 2L. r . = ' , x \ kz' cosh kz' +  (cosh 2k1i  a)sinh kz } • na "* n P K (where sinh 2i<h + 2k1i = 0)
K kA(cos1i 2k/i+ 1)  2 ' ) ft 1  k 2
with
P„ = J„,k/3 cos m«>(i<aG J kccJ ,„/3a — G m Kaj3aJ „,' fia) , if p<a
= G,„Kp cos 7)iw(ko,J ,„' kci3 Vl f$a — J,„Ka • (3aJ„'{2a) , if p>a
(66)
(ii) When /><«,, series which can be summed in finite terms,
<f)= 27TJ .„,/?/) cos too
y. smh KZ — , { kz sinh kz \{a. + cosh 2/Ji)cosh kz \ (where sinh 2kJi  2 K h = 0)
« (p 2  k 2 )k/i(cos1i 2«/i  1) (2 J
4. zL  '—7 ' h " ^ \ kz cosh kz' +  (cosh 2k1i  a)sinh kz' } (where sinh 2k// + 2k// = 0)
_ + «(yS 2  k 2 )k//(cos1, 2,<h + 1)1 2 ' J v
(G7)
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 171
In order to sum these last series, consider the function of k }
cosh k(z — z) / . \
+ o , n« w~ ( ± according as j> or <z )
2k{[J~k~) \ ° J
sinh kz ( , . , 1 \
sinh kz (" , , i
+ (FK>) K (smh2 K h2Kh) \ KZ smh KZ  K« + C03h 2 ^')cosh kz' J.
cosh kz ( , , . , )
+ 77w g\ / • t. n / n~~ iv •! KZ cosh K2 + A(cosn 2kA — a)smh kz > . . . (68)
(p~  K)K(simi 2k/i, + 2k/i) \  v ' j v '
Looking back at (61) we assure ourselves that this function vanishes at infinity in
such a way as to make the sum of its residues zero. Also, since the function is odd in
>c, the residues at k= ± k 1 are equal, and therefore the sum of the residues at the zeroes
of sinh 2kJi ± 2kI% is simply the coefficient of 27rJ m /3 j o cos mu> in (67). The sum of the
residues at k = ± /3 is
cosh f$(z  z)
+ w~
sinh f3z
MsinhVph 2(3h) { ^ sinh ^ ~~ ~^ a + C08h W cosh P z '
cosh /?z
2 (sinh 2(3h + 2j3h)
i /3z cosh flz + £(cosh 2/3A  a)sinh /3z' I . . (69)
A P
If this last expression near /3 = be of the form ^ + 5 + . . . , the residue at k = q
of (68) is simply  j^  p .
Hence the coefficient of 2TrJ m fipcosmw in (67) is simply (69) with sign changed and the
terms of negative degree in /3 subtracted. These terms of negative degree, just as in
§ 12, are added on again when we take in the part of the solution coming from (65),
which is obtained by writing F for x in (65) where
F = II x(R)J m /3p' cos niw p dp' dm.
2tt
— "04 Jm/^P cos mm + Fi • (Introd. (/).)
The term ~^ m fip cos mw being taken in for the purpose just mentioned, we are left with
F : instead of x in (65). Since y 4 Fi = , these equations now define a combination of
deformations of the persistent or permanent type, under no body force and no surface
traction.
The solution therefore resolves itself into
(i) this free deformation of the permanent mode ;
(ii) a local perturbation ;
(iii) a particular solution giving the proper discontinuity of stress correspond
ing to the applied areal force.
172 MR JOHN DOUOALL ON AN ANALYTICAL THEORY OF
The particular solution is
 2 cosh /?(z  z)
\ ± 2/3
^ = 27rJ„,/3/)cosm„» J sinh/fe
Mjd pr= mP z ' smh * " > a + cosh m cosh *'>
+ ^sinhtA 2^) ( ^' cosh ^ + * c0 ^~ 2 /^ sinh &
(70)
with a corresponding expression for d, obtainable from (61) by changing k into /3 and
then replacing J /3B by g '2*3 J* p cosmw .
It is easy to verify that this is actually a particular solution. Consider in the first
place the analogous forms of <p , in (61), and for greater generality, suppose J kR
replaced by f(x, y) where ( V 2 + * 2 )f= .
Then, from the method by which (61) were found, it is obvious that they give no
stress across the planes z — ± h . Let us examine the effect of the discontinuity in the
forms of <p , at the plane z = z , on the displacements and stresses as given in (4), (5).
If we take simply
= g cosh k(z  z')f
— a
cosh k(z  z')j—z sinh k(z  z')J
then we find at z = z\
u — v = w =
;.r = nj =
S = //(a + l)*/'.
Thus with the complete expression (61), the displacements are continuous, as also the
stresses z7, zy, but ^ (z = z + ) exceeds » (2 = «'  ) by  2m(« + 1)*/*. We thus see that
in (70) the corresponding discontinuity in 7z will be — 4x/<(a + 1) J m {3p cos ma> . This
continuity of displacement, and discontinuity in zz, are precisely as demanded by the
conditions of equilibrium of the plate.
If we take (61) with J kR unaltered, prepare them for integration as in § 19, multiply
by e' Ke and integrate with respect to k from to 00 , the discontinuity in zz at z = z' will
become
2fi(a+l) e« K,J KRd,K .
If further we multiply this by Z(x', y', z')dx' dy' and integrate with respect to x', y',
and then take the limit for e = 0, the discontinuity becomes, in virtue of (11),
brfi(a+l)Z(x,y,z')..
We have thus a proof of the solution for an areal distribution of Z force, independent of
the infinite solid solution (6), which might itself be found from the beginning by this
method.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 173
22. The general problem of internal normal force. Approximate forms
of displacements and stresses.
The developments given in §§ 14, 15 may obviously be applied in the present case
also.
Thus if in (61) we divide by kJ kR, expand in ascending powers of k, put — V 2 , that
T2 + j— 2 ) f° r K% i anf l operate on Z (x, y, z'), we obtain a form of solution which,
with the interpretation of V ~ 4 Z and V~ 2 Z given in (38), is simply the foregoing general
solution for areal force of intensity 2m(«+ 1)Z , arranged in terms of ascending order in
A 2 . The solution in this form fails if at (x, y, z), Z or any of its successive derivatives
become discontinuous, but it has been shown in § 14 how the local perturbation in the
neighbourhood of any surface of discontinuity may be calculated.
For the case when Z vanishes outside an area A, the principal part of the perturba
tion at the edge of A, when h is small, is found by substituting for G kR in (63), (64),
^ j { Z(Y, y') ^ G kR  G (*R)^ %&> //) J ds ,
where differentiations and integrations have reference to the accented coordinates.
Since the solution for the case when there are any finite number of surfaces at which
Z or its derivatives become discontinuous can be found from this elementary case by
simple summation, we see that discontinuity in the force itself gives rise to values of <j>, 6
in the perturbation terms of order h 2 at the surface, discontinuity in j to terms of order
h s if Z itself is continuous. The next term is of order A 4 and depends on discontinuity
of V 2 Z , that is, of the second derivatives of Z, and so on.
The symbolical solution for Z force distributed on the plane z = z with intensity
2/u(a + l)Z(cc, y, z') per unit area at (x, y, z / ) is given by
<f> = ± ^ cos1 ' K ( :: ~ "')
sinh kz ( , . , , \
7\ [kz sinh k,:  \a + cosh '2k/i cosh kz \
^ K 2 (sinh 2k7i  2k/;)
cosh kz ( . , ,\
+ ~ K \smh 2 K h + 2Kh \ KZ Cosh KZ + i cosh 2Kh  a smh KZ )
a _ Z
+ ^— 5 cosh k(z  z) +  sinh k(z — z )
o/ • L » i u \ I  kz' cosh '2k1i sinh kz + ~ cosh 2kIl + h + 2k 2 Ii 2 cosh kz)
« z (smh 2i<h  2kIi \ 2 * J
cosh kz
+  9/ ■ r . 7 , o i.x ( kz cosh 2kK cosh kz + \   cosh 2kIi+ 2k//. sinh kz )
K 2 (sinh 2k/i + 2k1i) \ 2 /
with k 2 = v 2 , operating on Z(x, y, z') . . . . (71)
The approximate solution is obtained by expanding in ascending powers of k 2 . By
retaining only the terms of negative degree in k 2 , each of the displacements will be
17 1 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
given to a second approximation, and each of the stresses, except ze, to a first approxi
mation. The result is obviously the same as that found by integrating the permanent
terms (65) of the original source solution.
If we write
T? = jfz(x',y',z')x(R)dx'dy',
then the displacements for Z force of intensity Z(x, y, z') per unit area on z = z' are
d ,
It. 3 / .\ti „,, /a+5,,a — 3/o +1 ,, a — 3 79/
 (a + 1),~F + v "* s v + .  2 z  Z hh + —— hh
dy
d SJhrfA 8  v ' v I 6 2 5 3
^ { (. + I)F + <*£?*+ ^W + if 1 V) 
32^ ' ; ■' ' : ( '
The corresponding results for a volume distribution of force, Z(x, y, z) per unit
volume, are found by integrating these with respect to z' from — h to + h.
In order to calculate the stress zH from displacements, we should need the value of w
to a third approximation. It is therefore easier to find zz directly from (71) and (5).
On dividing by 2n(a + 1), we find, corresponding to (72),
37i 2 z  z l
£=*iZ + — mT Z ■ ( 73 )
When the force is Z(x, y, z) per unit volume, this leads to
zz= 4 p J Z(x,y,z)dz J Z(x,y,z)dz+^JZ(x,y,z)dz . (74)
We can now find the stresses xx, xy, yy to a second approximation.
For
il x ihj dz
 v du , .dv ,v , .cto
" = A dx + % +{X+2fl hz>
whence, eliminating c ^° ,
° «z
~ 4:ii(X. + u)fdu , cfoA A _
as« = V: :V I — + °"— ) + ; , zz,
X + 2/j \dx dyj X + 2
and similarly
Also
* y==t \dy + Tx)
We have now only to put in the values of u, v from (72) and the value of zz from (73).
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 175
3
If we denote ,Ja + l)F, the principal term in w in (72), by W, we have
'o+l)Z
16/*/;
3 X + 2/x z
8/x/t 3 A + /x
or Cv 4 W=Z, where C =  A i/i 3 (A. + / a/(A+ 2/x) . . . (75)
In this notation, to a first approximation
~ = _ 3C_ f d 2 W d 2 W
°° x 2hf\ dx> d~y*
3C / rPW <fW
2hf\ dip If
yy= rf°" tzt 4
3C M W <PW ,„.
Again from (72),
Vcfe (&k/ 32irfj.h 3
= i C ^^^V 2 W . (77)
23. Normal force a function of z onfo/.
It may be useful to put down here the next term in the development of w, of which
the two principal terms are given in (72). This is
327rfxh
V 4 F
24 l ^ ; + 4(a+l) + V 10 a + l/ l ' \8a+3 5 + 352.7 /
(3 — a) 2 , o ,
3a + 3
(78)
±(la)(zz') Z(z,y,B')
In this, of course,
V 4 F = 2ttZ(x, y, r/).
The terms which have to be added to (72), (78) in order to give the complete particular
values of u, v, w, all contain x, y derivatives of V 4 F or Z. Hence, if Z(x, y, z') is a
function of z' alone, (72) and (78) give a complete particular solution of the problem.
Further, Z may have one constant value in one region of the plane z = z', and another
constant value in another region of that plane. (72), (78) will still give a particular
solution in each of those regions taken separately, or rather in the cylindrical spaces
of which these regions are sections, but it ought to be carefully noticed that it is not
in general an exact solution when the two regions are considered together as part of
one body. The point of failure is, it need scarcely be said, the condition of synexis ;
176
MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF
the two particular solutions do not fit, that is, they do not give the same values for
displacements ami strains on the two sides of the cylindrical surface or surfaces of
discontinuity.
On the other hand, the supplementary terms required in order to make the solution
synectic belong to what we have called the decaying type. They give rise to displace
ments and strains of infinitely high order, if we may so speak, in the small quantity h,
except very near the surfaces of discontinuity. This being so, we need not be
surprised to find that the solution (72), (78) is not necessarily the simplest particular
solution in any one region within which Z is continuously constant.
Thus, for example, if we pick out the terms which contain z' 2 as a factor, we find
displacements proportional to
*V 2 F
w =
a_
dx
dy
v 2F +v 4 f(^^ + c^
which belong to the type (23), and contribute nothing to body force or face tractions.
These terms might therefore be omitted in any problem where the condition of synexis
is irrelevant, and in particular when the object is merely to obtain a particular solution
for body force and face traction in a problem relating to a finite solid.
24. Internal force 'parallel to the faces.
We will now go on to consider the problem of force applied to the body in a
direction parallel to the faces of the plate.
A force of 47r / «(a + 1 ) units applied at (x f , y r , z') in the direction of 0.x gives in an
infinite solid displacements defined, according to (9), by
>A =
a+l d
2 dy dz~ 2
a d d' 2 r~ l ,d d~ : r~ l
_ + z —
2 dx dz"' 1 dx dz~ x
1 d d 2 r~ l
2 dx dz 2
Hence the tractions which such a force produces on z = ± h will be neutralised by a
system \f/ , 9 , </> for which
d\jr _ d fa. + 1 d~h'~ l
dz dy\ 2 dz~ l
d A + !*£ + 2z d2< t = d ( a + 1 d ~ lr ~ l
dz dz dz 2 dx\ 2 ~dz zr +
iP6_d 2 cf> :) d 3 <f>_ d/al . dr x \
dz* dz 2 dz* riA~2~ r +z ~ z ~dJ)
on z= ±h .
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
These conditions are satisfied if
and
We may take
</'
dy'
dz
dz
dip _ a + 1 d
dz
dz
dff d<b' dU' a+1 d~ l r~ l ,
d 2 6' d*d>' d s d>' a1 ,
dz 2 cfe 2 + Zz dz* ~ 2 r
+ Z  2
<2r'
tfz
jy 1
dz~ 2
dry
dz
= P ' j eT^^J^R _ 1 + K ( z  jQfli* J ™
_^=^( + ) je+^OJ^R
e+^m k R^k
177
(79)
upper or lower signs being taken, as all along, according as z > z' or z < z' . We therefore
determine provisional values of \//, 0', <£', such that
d\l/' a + 1 _ , , „
d 2 6' d 2 d>' d z <h' fa  1 A ,
W  W +2z l§ = {2 + KZ * )^(^)J„kE
These provisional values are easily found to be
^r' = C ^ r COSh/c(22')
a+1
2k 2 cosh xh
a+1
2/c 2 sinh kJi
sinh kJi sinh kz' sinh «2 V J kR
cosh kJi cosh k2' cosh kz
$ = — 2 cosh k(z  z)
sinh k2
— <  kz' cosh kz' + — (cosh 2kA  a) sinh kz' >
IkIi) I 2 J
K 2 (sinh 2k/i  2k/j)
cosh kz ( i I
—: — —  < kz sinh kz' + — (cosh 2k7i + a) cosh kz }
2kJi + 2kA) I 2 )
K 2 (sinh :
J kR
9' = — 2 cosh k(z z)  — sinh k(zz)
2k' k
+
.. . ? r 7 KZ „ , , < kz' cosh 2k/i cosh kz' + — (a cosh 2k/i  1  4k 2 7i 2 ) sinh kz >
K 2 (sinh 2k7i  2kJi) I 2 J
„ . , „ , KZ 7T I kz cosh 2k7i sinh kz + — (a cosh 2k/i + 1 + 4k 2 /V') cosh kz' >
K"(smh 2K/i + 2K/i) ( 2 _L
TRANS. ROV. SOC. EDIN., VOL. XLI. PART I. (NO. 8).
JqkR
(80)
28
17!
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
The source itself is similarly given by the temporary values
r
a+ 1
Ik k
> c=F«(«OJ kR
~ o '1
Thus, when the source is included, the provisional values of \/', &, (j)' are as in (80), but
with the first lines altered,
in i// to ± „ sinh k(z — z)
2k'
„ <f>' „ ± —. , sinh k(z  z)
Ak
„ & „ ± ss sinh k(zz) + — cosh k(zz')
2k k
I <LkR
(81)
25. Solution of the problem of internal force parallel to the faces.
From these expressions the solution in the form of definite integrals, and finally of
series, is obtained as in the previous cases. After the explanations already given, it
will be sufficient to write down the final results. For the transitory part of the
solution ,
\p' = 2j (a + 1) — sinh kz sinh kzG kR , (k a pos. imag. root of cosh kK)
k K ll
 2i (a + 1)_ cosh kz cosh kzG kR , (k a pos. imag. root of sinh kTx) .
_ 'V sinh kz G^kR
K I
= 2a same as previous line multiplied by (  cosh 2kJi)
4> — 2i „ ^"i^l^nf^L ) _ KZ ' gosh kZ ' _j_ (cosh 2k1i  a) sinh kz' \
k/« (cosh 2kK 1) I 2 j
where k is a zero of sinh 2di — 2k1i, with pos. imag. part.
With
'=£<),
$=
cosh KzG n /<R
k 2 A (cosh 2k1x + ] )
< kz sinh kz' + — (cosh 2kK + a) cosh kz >
6' = 2i same as previous line multiplied by cosh 2k1i
K
where «■ is a zero of sinh 2kJi + 2>ch, with pos. imag. part
(82)
We may recall the method of obtaining the permanent terms. Taking any one of
the functions of (80), altered as in (81), we omit the factor J a:R,, and then find its
expansion near k = to contain terms of negative degree in *, say A/k s + B/k. The
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 179
permanent part of this function is then Ax(R)  By 2 x(R) We tnus nncl for tnis P art
of the solution
3(o + 1) , / 1 3 2 \ . 3 (a + 5 , 3 } , , _ a + L v \.
_ 1 / ,V ,
+ y(22)V'X
, 3(a+l) ,/ 1 32 o; , . \ 3 /a + 5, 3 j„, a+LjA
+ ^F^ H 2X " T 2 V " X  27r2V " X ) " 4/?^ ' k ' Z ~ W h Z > VX
(83)
When z' is put equal to h in the above values of \f/, <//, 0' it will be found, with
very little trouble, that they reduce to those of (48) .... (51), multiplied by (a+ 1).
{Of. § 20.) As in § 20, the displacements due to the ambiguous terms in (83) are null
if R > 0. But there is this difference in the present case, that they do not continue to
vanish in the corresponding solution for an areal distribution of force onz = z'.
If the intensity of the distribution is X(x, y, z') per unit area at (x, y), this solution
is defined as in (79), \//, 0', <fi being obtained from (82), (83) by multiplying by
and integrating over the area within which X is finite.
When this is done we find that the ambiguous terms lead to
u= +—(zz')X
•Ifx.
v =
w =
(84)
In verification, we observe that these displacements are continuous above and below
the plane z — z', and that the corresponding stresses are also continuous with the excep
tion of zx, the value of which just below z = z' exceeds its value just above by X. The
value of «e being TfX, we have for the contribution of (84) to the resultant J zx dz,
yX(f dz j\foWx . . •  • (85)
with
180 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
26. Approximate values of the displacements. Lagrange s equation for flexure
to a second approximation.
The unambiguous terms in (83), as in (82), fall naturally into two classes, in the
first of which >//, 6', ft are odd functions of z, while in the second they are even. Of the
displacements derived from the first class, u and v are odd, and iv even in z, and the
strain may be described as flcxural. In the other class u, v are even, and w odd in z,
and the strain may be described as extensional. A force X at (x f , y', z') acting along
with a parallel but oppositely directed force X at (x', y', — z') would give rise to flexural
strain only ; equal and similarly directed X forces at these two points to extensional
strain only. This follows at once from the fact that the terms of \J/, 6 r , <fi', which are
odd in z, are also odd in '/, and vice versa.
The distribution of force being X.(x, y, z') per unit area at (x, y) on z = z', let
Y = jLjjX(z',y',z') x (R)dx'dy'.
Then from the flexural part of (83),
*"H5P'(*M
<£=£V''F . I4 _.,. ,, 3 2 /o + ll, v a + 5 ,A
e= SV , F f each multiplied by 32 — K , • ^jQ *"* ~ 12 z 3 )
These lead to
_ d
dy I
For Y force the same expressions hold if we take F = ^ M Y(*', y', 2') x(R)^*W
These formulae, with all of (72) but the last terms of u, v, and with the odd parts
in z arising from the ambiguous terms, give to a second approximation the displace
ments of the flexural mode under any forces. The differential equation satisfied by w,
the normal displacement of the mid plane, or value of w, for z equal to zero, is important
in the history of the approximate theory. We can now write it down to a second
approximation, namely, with C as in (75),
cv^ = z + (^ + ^)
V dx dyJ
(a 19 h 3<xz' 2 \ 9 „ /.,., a + 5z' z \ /dX dY \
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
181
This equation gives the result for an areal distribution X , Y , Z on z=z' . For
traction on z = ±h replace z' by ± h ; for a volume distribution X , Y , Z replace
Z by i K _Z(x,y,it)dz'
,/dX dY\ , d f> ,„. ,.,, d (x
and so on.
27. Extensionai strain. Differential equations of the 'principal mode.
The unambiguous extensionai terms of (83) remain to be considered.
Write E = 3 2 J jX(x',y',z')x(B,)dx 7 dy'.
Then for an areal distribution X , these terms are
$=  4(E  £*VE) + 4(iz' 2 + ^ 2 )v 2 E
2 / a + 5 ,.. 5
' (E^v'E) + ^^ + ^ +
a+l/r J V 2 E
2 / a+5 ,, 5a„ \ „_
+' =  (E  ! 2 VE) + j (  JW + 6 /r J V 2 E
The second parts of these expressions give
; = (a  3) (J/ 2  J ft 2 ) ^ 2 V 2 E + 4(z' 2 + §/r) V ' E
J 2
w; =
E
The first parts give
u
d 2 E Q d 2 E , o /a 3 rP
. g tZ 2 E , 2
^ a ' «te dy dx dv
d*E_
dy
2 dx 2
a  3 tf 2
2 dxdy
V 2 E + 4v 4 eV
V 2 E
d
w
= (3 a) ^ V 2 E
<ia;
If now further we write K = ^^ / j Y(tc', ?/', z') x (B,)dx'dy'
the corresponding displacements for a distribution of Y force on z = z' can at once be
written down from symmetry. The results for X and Y force combined cannot con
veniently be expressed in terms of one function, as in the case of the flexural mode,
and the best plan is probably to put everything in terms of the principal values of
u , v , namely,
U= (o + l)
^ 2 E „rf 2 E , . 1N d 2 K . a d 2 K]
dx 2
dy 2 dx dy dx dy I
y ( a + i)^ 2E +8 <* 2 E _/ a+1) ^K _ 8 d 2 K\
dxdy dxdy dy 2 dx 2 J
(88)
182
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
We have then
X ^
fxh
g3 ,,/<*£■ i*y
a + 1 ' \ flte ay
(89)
The ordinary approximate theory obtains differential equations to determine
U , V . These are easily found by eliminating E and K in turn from (88).
Thus
dTJ dV _ _, ,i\(d 2T£ + d 2 K \
dx dy \dx dy J
dy dx \dy dx I
and
1 d_(dV dV\ ,idfd]J _dV\ X ^
a+ 1 tfaA cfo <ivy / c?//\ (/?/ da / [6/j.fi
_J_ d/dV dV\_ 1 d_fdV _dj) = Y
a+ 1 ayl 'te ^/7/ / dx\ dy dx ' \QfihJ
(90)
The principal parts of the contribution of Z force to extensional displacements
appear in (72), (78) . In the notation of those formulse
dx
', = /v 2 F
dy
a+ 1
2V 4 F
327r/x/i
with in addition, w = the odd part in z of the ambiguous term in (78).
If these last values of u, v be included in the principal values U, V, then the right
hand members of (90) will become respectively
1
1
a — 3 ,dZ \ 1 / a — 3 ,dZ
a+1 ax j , lbfjJi\ a+1 ay
(91)
28. Approximate values of the stresses across a plane parallel to the faces.
For any distribution of force parallel to the faces of the plate, the formulae of
§§ 26, 27 give the terms of the two lowest orders in the values of u, v, and the term
of lowest order in w* From these terms we can calculate all the stresses but S to a
first approximation, and as in § 22, when the first term of 7z is known, we can find
two terms of xx, xij and yy. This first term of zz we may get very easily from the
symbolical form of the solution corresponding to (80), (81). Thus for areal force X
£= ±W ?: )
dX 3 ,„. , R dX 1 ., ., dX
zV™\*)liZ T h (* 2 + h)
~dx + 4/t 3
dx ih x
dx
(92)
* There should be added from (84) the terms
,' = T(>~')X/2 M) «=T(s« , )Y/2/ tl '" = 0.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 183
From (84), (86), (89) for areal force X
3 ,,.. l d 2
sy
+ hX + ^ z'(?  h*) ¥i? ^ J jX(x', y, z) y \(^)dx'dy' + ~X '
W ZV ~ h *)*r didyj j X ^'' ^ OVxCB)^'^
(93)
"zr «zy "2
It may be verified that these give zero stress onz=± /i, and v + p + 5 = 0.
From the formulae we have given, it is of course merely a matter of the simplest
algebra to calculate any of the stresses to whatever order of approximation may be
required, but it may be worth while to remark here that the fundamental equations
of equilibrium (1) may be used with great advantage in obtaining the principal results.
If. for example, we know only the first terms of xx, xy, yy, the two first of these
equations would give the first terms of zx, zy by a simple integration with respect to z,
and then the last equation would give the first term of zz. Similarly, when the first
two terms of xx, xy, yy are known (as above), we may find the first two terms of the
other stresses.
29. Transmission of force to a distance. Expansions in polar coordinates.
We have up to this point been considering mainly the particular solution to
which our general source solutions lead for any given distribution of force ; or, as we
may say, we have been investigating the effect of any given force system on that
part of the solid to which the force is applied. But it is also of great interest to
inquire what is the effect of this force at points of the solid remote from its region of
application. It is obvious that we obtain a sufficient answer to this question by
retaining only the permanent terms in the source solutions, those terms, namely,
which are given in (65) and (83).
For force applied only at points on a given normal to the plate, these formulae
are all that we require. They show at a glance that the distant effect depends chiefly
on resultant forces and couples, but not entirely, since z' and z n occur in the formulae
for Z force, and z n , z' z in those for X force. When the force is not confined to a line,
but is distributed over a finite volume of the solid, the result is obtained in more
intelligible form if before integration the function x is suitably expanded so as to
yield a series of solutions in which accented and unaccented coordinates are explicitly
separated. The most convenient expansion of x is m terms of polar coordinates as
given in (e) of the introductory section.
Suppose, then, a single force applied at the point (x x , y x , z r ) or (p x , w 1 , %), the
components of the force being X 1 , Y x , Z x , parallel to the rectangular axes, or P x , Q x , Z 1
parallel to radius vector, transverse, and axis of z. We have to find the displacements
at (p ,w ,z) where we suppose jo > p L .
For an X force, the value of ^ is ^ with \/ given in (83) , the coefficient de
pending on the magnitude of X being for the moment suppressed.
184
Ml! JOHN DOUGALL ON AN ANALYTICAL THEORY OF
This is the same as — ■$, or ( — ) rate of variation of \J/ in the direction perpen
dicular to the force at its point of application.
Similarly
Hence for
<£ = _ X _ ( _ ) rate of variation of <f>' in direction of force.
dx
1 cty'
* =
d Pl
and for O,
«P =
^Pi J
\dfi_
p, do)j
* = ~d^J
We shall take separately the extensional and flexural parts of the solution.
Also in the following u, v are the displacements along radius vector and transverse.
I. Extensional terms.
The following solutions occur.
[1  a. 9 + a, p 3  a 9 _„ \ s
(!) U = ( — r log ^ + jZ^p ) COS W
7a 9 + a. p 3 — a „ _,
2
u= (V + ^ l0g 4 +
M> = (3 — a) Zp~ x COS w
(ii) Same as (i) with cos w changed into sin w, and sin w into  cos w
(iii) When m>l,
; ^^2 n — l I COS mco
4(to1)
mz'p
.{
3,
m(a^_ m+1 + ^a m ^
2_ — m
il
4(ml)
M> = (3  a)zp""' COS TOW
(iv) Same as (iii) with cos mw changed into sin mw, and sin tow into  cos tow.
(y) U = p~ m ~ 1 COS TOW I
v = p m x sin ???w
(vi) ?/ = p m * sin tow
v= — p
m— 1
r?il
COS TOOJ
}
For the force with components F 1 ,Q 1 ,Z 1 , the coefficients of the above solutions
are the following, in each case divided by 32Trfj.h.
(i) Pj cos Wj  Oj sin Wj = X x
(ii) P x sin w x + AJ cos Wj = Yj
(iii) Pj"'" 1 cos TOw 1 P 1  P!'" 1 sin mw 1 1
(iv) Pi" 11 sin tow^ + p," 11 cos tow^,
(v) { "^ffiffi+V " + (3  ) (K  ^)P™ 1 } oo ^
+ { 8(m+ 4 g; + ^ a ^ r +1  (3  a) (^ 2  ^Wx" 1 } n mcoA
+ (a — 3)p 1 '"z 1 cos towjZj
(vi) Same as (v) with cos mwj changed into sin mw x , and sin mwj into  cos jbu,
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
185
II. Flexural terms.
The flexural solutions are of the form given in (23), or in polar coordinates
d
■ (a + l)(zF  \z? V 2 F) 4 2 (1 2 3 _ m) y 2 F
dp
1 d
p db
IV =
(a + 1)(F 122 y 2 F ) + 2(.; 2  fc») V 2 F
(94)
(i) F= x (p) = Vlog^ip
(ii) F = i( P 2plog)cosa
(iii) F = i^2 P log 2 y sin a
(iv> F =4^=rr
(vi)F = log£
> ?»■> 1 .
(vii) F = p '" cos mo> \
(viii) F = /j  "' sin ??iw '
m>0
For the force with components P x , Q x , Z x the coefficients of the above solutions are
the following, in each case divided by 4f tt/x/i 3 .
(i) Zi
(ii)  2 X cos ojjPj + 2j sin WjQj + Z^ cos a>j =  X^ + ZjSBj
(iii)  z x sin oj 1 P 1  z l cos WjQj + Z 1 p 1 sin w 1 =  YjZj + Z^
(iv) z^np^"^ 1 cos raojPj + ;j 1 ?»/3 1 '"~ 1 sin maj 1 fi 1 + pj m cos maijZj
(v) Same as (iv) with cos ?ri(o 1 changed to sin mwj , and sin mwj to  cos jhgjj
(vi)  \z lPl v x + { ^ + iv  &* + JL(% 2  **) I Zj
( a + 1 J
(viii) Same as (vii) with cos ma, changed to sin mo> x and sin mu> l to  cos vtu 1 .
30. Types of deformation conveying a given resultant stress.
In these formulge we remark at once a striking relation between the forms of the
displacements u, v, w in the various solutions, and the multipliers of P x , Q x , Z : in
the coefficients of the solutions.
In I. (iii), e.g., these multipliers are f) 1 '"" 1 cosma> 1 , — pj" 1 " 1 sin ?nco 1 , , which are
simply the displacements of I. (v) with sign of m changed, and consequently suitable
for space containing the origin.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 29
186 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Similarly in I. (v) the multipliers of P : , ^ x , Z l are displacements compounded of
the types (iii), (v), with sign of m changed, and so on.
The full explanation of this peculiarity will be given presently, when it will be
seen that an independent verification of all the results may be obtained by means of
the important principle known as Betti's Theorem.
In the meantime we may examine the scheme of solutions from another very
important point of view.
With reference to any individual solution, the following questions are obviously of
prime importance : —
( 1 ) What is the resultant stress transmitted ?
(2) Is the whole potential energy of the part of the solid bounded internally by a
given cylindrical surface, finite or infinite ?
Now, in order to single out those solutions which convey a finite resultant stress
across any cylinder (or other surface) surrounding the origin, we have merely to look
at the table of coefficients. Thus, for instance, I. (i) appears with coefficient X 1 /32Triu.h,
from which we may infer (as verified below) that this solution conveys a stress with
resultant a force of S'ln^h units parallel to the axis of x, and passing through the
origin.
In this way we find that the six solutions, corresponding to the six elements which
specify the resultant of a force system, are
I. (i), (ii), (vi) withm = 0; II. (i), (ii) , (iii).
For these we shall write down the values of the stresses p~p, jTo, ^i, the components
of the stress across the cylinder p — constant.
In all, of course, we have zz = 0, and in I. in addition zp = zZ = 0.
I. (i) f p = (^V 1 +T 3 z* P A cos w
p ^ /a+1  —  \ .
2 = ^ 2 p l +a3zp ■>) sin w
The resultant is a force along Ox, of magnitude
I I (^ cos w  p ^ sin u>)pda>dz , taken over the cylinder p ,
= ( a  H _ a+lV . . 2 h ■ 2y. =  32*/* .
I. (ii) 7? /a 15 . — . _ A . 1
O,
^ — p 1 + a  5 zp 6 ] sm
pai /a + J 75 „ _
~~n~p +ad Zp MCOSco
The resultant is a force along Oy of magnitude — S'lnfxh.
L(vi),«i«0. «o I  =
W = po>  Ipp .
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
187
The resultant is a couple in the plane xy, of magnitude 8tt^/?, and we observe that
the solution occurs with coefficient — Q^/StthJi.
II. The stresses in the general flexural solution (94) are
II. (i)
f = (a  3> V 2 F + P
2fl dp
pa, dv
2p. dp
• •
f = 2( z 27i2)v 2 F.
2/x dp j
PP _ a ~ 7 z \ a P a+1 ? _/ a + 5 ,3_
2fi 2 b 2h 4 " V 6
 2/i% V s '
pa, = U
^.= 2(^A»)pi
>
32
The resultant is a force along Oz, of magnitude  ~viih* .
II. (ii)
iS = I— ^ Zp  l COS o,  2^— 7?  2/^2 )p» COS CO
2p 2 r V 6 > H
pai
2p~
.+1
2(" a "t 5 2 3_27 t 2 2 V 3
£ = 2(z 2  7i 2 )p 2 cos co
2/j,
The resultant is a couple about Oy, of magnitude
I I {z(^ cos to  ^ sin (o)  p cos a^jp^cocfe, taken over the cylinder p,
32
3
II. (iii)
32 „
— 7rp/4 3
—  — 2 3 — 27A )p 3 sin to
pp 7 — a , .
££■ = —  — zp i sin to
2p 2
^ = ^lzp 1 cosco + 2f a ±^2 3  !
J£ = 2(* 2  A 2 )p 2 sin co .
2p
The resultant is a couple about Ox , of magnitude
I / { — 2 (pP s ^ n (0 + p<" cos w ) + P s i n w • p^}p f?co (7z =  — 7rp/< 3
(95)
31. Conditions for the existence of a solution with finite potential energy.
Elastic equivalence of statically equipollent loads.
The corresponding results for any distribution of body force, or of traction on the
faces of the plate, may be deduced at once from the above by integration with respect
to />!,«!, z 1 or p 1 , Wj with z 1 = dtzh.
If the region within which the force is applied be entirely enclosed by a cylinder
p = a, the results are valid for all points exterior to this cylinder.
188
Ml J JOHN DOUGALL ON AN ANALYTICAL THEORY OF
For a distribution of force of finite intensity per unit area or per unit volume, the
potential energy of that part of the solid within the cylinder is clearly finite. The
energy of the remaining part of the solid can be determined from the forms of § 29.
Now, the energy between the cylinders p = a, p=p is the integral of
taken over the belt of the cylinder p = p cut off by the plate, diminished by the
corresponding integral for p = a. Hence the condition of fmiteness of the whole
potential energy is simply that the value of the integral for the surface p = p tends to
zero as p tends to infinity. This condition is obviously satisfied by all the partial
solutions of § 29, except those which have been already singled out as conveying a finite
resultant stress. It is also satisfied by one of the latter class, namely, that which
conveys a couple in the plane of the plate.
Hence, when force is applied to a circumscribed portion of the solid, a solution giving
finite potential energy will exist provided the force either constitutes an equilibrating
system, or reduces to a couple in the plane of the plate. It does not follow, however,
even for an equilibrating application of force, that a solution will exist giving vanishing
displacements at infinity. We need only point to the solutions of § 29, II. (vi) and
(iv), (v) with to ±= 2. This being so, it may be of interest to write down a few more
details of those solutions which rank in importance next to the solutions of finite
resultant stress.
I. (iii) with to = 2.
u { 4p _1 + (3  a)z'7> 3 } cos 2a>
v = I  ^±V ] + (3  a),r> 3 } sin !
w = (3  a)zp~ COS 2w
pp_ )
I£= J (a  7)p" 2 + (a  3)3zV 4  cos 2c
2u ( )
{^V 2 + (a3)3 Z V 4  sil
pa) _
2/x
This solution occurs with coefficient (X 1 x 1 Y 1 y ] )/327riuh.
I. (iv) with to = 2 is obtained by writing sin 2», — cos 2u> for cos 2w, sin 2w in the
preceding, and the coefficient is (X 1 y 1 + Y 1 sc 1 )/32?r/*7j..
I. (v) with m = 0.
U = p
v =
w =
Z ' ^ I Coefficient = ■[ a + (X^ + \\ ft) + (a  3)z 1 Z 1 \ l32vfih.
pa> — U I (2 j I
II. (iv) with to = 2. F =  cos 2«.
u — [ ( .' z 3  2/r.':;)p _3 cos 2w
a+ 1
a +
££ = —^—zp  cos 2oj + a term in p 4
■Zp. 2
o+l
v= A zp~ x sin 2to + l^z*  2hh V" 3 sin 2 W . ^ =  ^^zp' 1 sin 2 W +
\ 6 / 2u 4
a + 1 , (a — 3 ., ,
w =  . cos _'o) + I z + a ) p  cos 2w
^ = 2(z'  Jr)p 3 cos 2(
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 189
The coefficient is
II. (v) with m = 2 is the above with sin 2«, — cos 2w for cos 2«, sin 2« and
coefficient
II. (vi). F = log^
327^
24 =  (a 4 l)zp '
V =0
!0 = (a+l)log^
f =(a+l>p
pw  p.:
:o. )
Coefficient is
{ ^X^Y^ + ^^V^
For all the remaining solutions, the stresses are of the third or higher order in I//),
The results of this and the preceding article bear directly upon a principle of
fundamental importance in theories of approximation, generally referred to as the
principle of the elastic equivalence of statically equipollent systems of load, and
a study of these results will be found of service in imparting precision and
definiteness to one's view of the principle in its application to the theory of plates.
It may be noted here, with reference to the occurrence of the function log (p/2Ji)
in some of the principal solutions of § 30, that it would make no essential difference
if this function were replaced throughout by log (p/'c), c being any length whatever,
the unit of length for example. The change would be equivalent to adding a solution
of the permanent type, giving no body force or traction on the faces, and it will
be observed that the addition would disappear altogether when the applied forces
are in equilibrium.
We have here, in fact, an instance of the indeterminateness that of necessitv
arises in the absence of conditions at infinity, and we are thus brought to the
question, what is the exact extent of this indeterminateness ? or, as it may be put,
given one solution of a problem satisfying the conditions at a finite distance, what
is the most general solution satisfying such conditions ?
For the investigation of this question we have at hand a powerful instrument
in Betti's Theorem, which occupies in the theory of elastic solids the place held
by Green's Theorem in the Theory of the Potential.
32. Betti's reciprocal theorem. Verif cation of preceding solutions.
Betti's Theorem may be thus stated : — Given two sets of displacements of an
elastic solid, with the two corresponding sets of forces maintaining these displacements
(including body forces, surface tractions, and kinetic reactions), then the work done
by the forces of the first set acting on the displacements of the second set is
190 MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF
equal to the work done by the forces of the second set acting over the displacements
of the first.
In potential theory one of the chief applications of Green's Theorem is to the case
when one of the potential sj^stems includes a mass concentrated at a single point, and
in the present subject Betti's Theorem finds an application of like importance when
one of the displacement systems contains a finite force applied at one point, or, in
analytical language, includes a point singularity of the first order, that is to say, of one
of the three forms indicated in (6).
Thus, let us suppose the solid to be bounded by a surface S, and in the first set let
the displacements be u, v, w ; the components of body force per unit volume X , Y , Z ;
and the components of the traction on S , F , G , H ; in the second set let the displace
ments be u r , v f , w' ; the only internal force a force X.',Y', 71 at (x', y', z'), and the
tractions on S , F, G', H'.
We may apply Betti's Theorem to the space bounded by S and a sphere S' of radius
e drawn round (x', y', z') as centre. Thus we have
f f f(Xu + Yv' + Zw')dY + j j (Fu + Go' + Hw>iS + f [{Fu' + Gv + Kw')dS'
= f f(F'u + G'v + Kw)dS + f f(F'u + G'v + H'w)dS' .
Now take the limits of both members of this equality for e = 0.
Since near the centre of the sphere S', u', v', w' are of order l/e, F', G', H' of order
1/e 2 , and cZS' of order e 2 , the effect on the volume integral is simply to extend it to the
whole volume within S ; the surface integral J j (Fu' + Gv' + H.v/)dB' vanishes, and the
surface integral I I (F'u + G'v + Ww)d& has the same limit as
u(x, y, z')JJF'dS' + v(x, y', z) J JG'dS' + w(x', y', z')jfu.'dS',
namely,
u(x, y', z')X + v(x', y\ z')T + w(x, y', z')7J,
the tractions F', G', H' on S' being statically equivalent to the force X', Y', 71 at its
centre.
It is thus apparent, and might indeed have been anticipated, that Betti's Theorem
may legitimately be applied when one of the systems contains a force acting at a single
point, provided the work done by this force on the other system of displacements be
taken into account.
The theorem thus becomes
f f f(Xu + Yv + Zw')dY + f f(Fu' + Gv' + Kw')d$>  f f(F'u + G'v + K'iv)dS
= u{x, y', z')X' + v(x, y, z')Y' + w(x, y, z')Z' . . . (96)
In order to apply the theorem to the plate problems under discussion, take for the
solid a portion of the plate bounded externally by any orthogonal cylinder. Let us
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 191
also suppose that the system u , v , w is maintained solely by tractions on the cylin
drical edge, and the system u', v' ', w' by such tractions along with the force at (x\ y r , z').
Further, it will be convenient to decompose the latter system, and take u x , v x , w 1 as due
to a unit X force, u. 2 , v 2 , w 2 to a unit Y force, and u s , v 3 , iv s to a unit Z force. The
corresponding tractions on the edge we will denote by X , Y , Z ; X x , Y x , Z x ;
X 2 , Y. z , Z 2 ; X 3 , Y 3 , Z 3 . The theorem (96) then gives
(97)
u(x, y\ z) — I I (Xm x + Yv x + Zw 1  XjM  Y x u  Zjw) JS N
v(x', y ', 2) = I j(Xu 2 + Yv; 2 + Zw 2  X 2 m  Y 2 vZ 2 w)dS  .
w(x, if, z) = I I (Xm 3 + Yv., + Zw a  X 3 u  Y. d v  Z 3 w)c£S
the integrals being taken over the edge.
As one application of these forms, we may indicate briefly how they can be used to
verify the single force solutions already obtained.
Take, for example, the case of a Z force, and let u s , v 3 , w s have the values defined in
(63), (64), (65). Also let the edge be the cylinder R = constant.
(i) The coefficient of the principal flexural term, in which, with the notation of (94)
Foox(R), is determined from the condition that the resultant of the stress zR must
balance the applied force.
It is interesting to note that the conditions of equilibrium of applied forces and
surface tractions may be regarded as special cases of Betti's Theorem. We have only
to take for auxiliary systems the rigid body displacements u = 0,v = 0,w=l; u = y ,
v= —x, iv = , etc.
(ii) In the third of equations (97) take for u , v , iv the values of (94) with F = R 2 .
Only the two flexural terms of (65) contribute to the surface integral ; the contribution
from the particular solution <p = G kR sinh kz , 9= —cosh 2kJi'<P must vanish, as we see
by pushing the edge to infinity.
This, with the result of (i), gives the coefficient of the second flexural term of (65).
(iii) The principal extensional term is verified by taking
a+1,. _,, a+1,
9,
. (xx), v= (yy'), w = (a3)z,
(iv) The coefficient of the particular solution <p = G kR sinh kz , 9= — cosh 2Kh'(p in
(63) is verified by taking for u, v, iv the values defined by (p = J kR sinh kz ,
9 = — cosh 2kJi '(p.
None of the solutions corresponding to the other roots of sinh 2kJi — 2kJi contribute
to the surface integral. In fact, the partial contribution from a root k' being inde
pendent of the radius of the cylinder, must vanish identically, since the Bessel Functions
supply a factor tending to zero or infinity when R is made infinite, according as k' is a
higher or lower root than k.
(v) The coefficient of the particular solution <p — Q k~R cosh kz , 9 = cosh 2kJi'(J), may
be verified in the same way.
;l
l'.)2 MB JOHN DOUGALL ON AN ANALYTICAL THEORY OF
It is now easy to see the significance of the forms of the coefficients in the solutions
of § 29 and the confirmation of the values there given would obviously present no
difficulties.
33. Finite plate under edge tractions. Form of the solution deduced
by means of Betti's Theorem.
We pass, however, to a more important application of the theorems (97). The
system u , v , iv we still suppose maintained by edge tractions alone, but in addition to
the external edge the solid may now be bounded by one or more internal edges. For
"i » v i , Wi, etc., we take the definite values defined in (79), (82), (83), and in (63),
(64), (65).
Thus in (97) u x , v 1 , Wy , X x , Y x , Z x , and the other displacements and tractions marked
with suffixes, are known functions of x', y', z', and the equations give explicitly the
values of the displacements at any internal point in terms of the displacement and
stress at the edge or edges.
The ideal solution would give the internal displacement in terms of edge displace
ment alone, or of edge stress alone, but the analytical difficulties are such that we are
unable to solve the problem thus completely even for the simplest case, that of a single
infinite plane edge. Meantime, however, we may derive valuable information from
the expressions of (97), and in the first place as to the form into which any solution
due to edge tractions alone may be thrown.
Just as in the case of the original source solutions, we find that the solution, in
which, of course, the accented letters are now the variables, may be decomposed into an
extensional and a flexural part, while in each of those parts we may separate a permanent
mode from an infinite series of transitory or decaying modes of two types, the \J/ type,
characterised by no dilatation or normal displacement, and the , (p type, in which
there is no molecular rotation in the plane of the plate.
In the following analysis integrals of the same form as those in (97) occur
frequently ; the system u , v , w appearing in each case, but associated with various
other systems. For conciseness we shall refer to the first integral of (97) as the work
differ owe from Uy , i\ , Wy , and similarly in other cases.
I. Extensional part of the solution.
(i) Permanent mode.
In Uy , vy , Wy , the terms which relate to this mode are the unambiguous terms, even
in z, of (83), after these have been divided by 47r,u(a + 1). These, as may be seen from
a glance at the beginning of § 27, are equivalent to
. _J_ d_f _J 2 .
^ ' Srr/Ji r/y'V* 2 " V X
1 / d ( 1,.,\ 3a/l , 1 . 2 \ d ,,
32nfj.J/
V* standing for JU^ 2
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
193
Now let the work difference from the system >A = g — Jx  j z2 v 2 x) be denoted by E x , and
that from the system = <f> = — — Jy  y*V 2 x ) by E 2 ; then obviously the work differences
from the two systems immediately preceding are respectively
^i and —f + 3 "Y 1 e f '  — ^A— V ' 2 E .
<#?/' <ta' a+l\2 3 /eke' 2 '
Hence (97) gives
In the same way from w 2 , ^ , w 2 and w 3 , v 3 , w 3 we obtain
V(X V z)   f ' E l + dK 2 + 3  a f 1 /2_ 1 7, 2 \ <* ^'21?
w(a y ,z) = 2 y Z E,
a+ 1
Moreover, it can be seen in a moment that the displacements due to
<A = r,v" 2 x and t0 = <£ =  — ^ 4' v 2 x
a# af 1 dx
are in reality the same ; as also those due to
It follows that
i^ = — — ,v' 2 x and to 6 = d> = ^— — y' 2 y
r dx A a+1 chj X
dy a+1 dx
and / /V '2 El A_ 4v'% =
dx a+1 dy ' J
If we write U for
and V for
dE l (IE 2 _3a Id , 2F
dy' dx' a+1 J dx~' V 2 '
(iEj c?E, 3  a 1, 2 d , 2 tj,
efo' (/// a+1 3 oft/' 2 '
we obtain the form which it is convenient to take as the standard for this kind of strain,
namely,
.,/nt ./ j\ tt , 3o 1 ,„d fdU dV\)
u(x,y,z) = V + z 2 +
a + 1 '1 dx \ax ay J
«.m v + .m^^4> <93)
with
a+1 Veto (/// /
dy\dy dx) a + 1 dx'\dx' + dy) ~
d_(dU _ dY\ _ _8__ d_A7U d_V\ _
dx'\dy' dx) a+1 dy\dx' + dy')~
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8).
30
194
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
(ii) Transitory modes, \f/ or rotational type.
Referring to the expression for \J/ in (82), put
E 3 = work difference from the system ^ = g — ^ cosn k^o" 5, •
Then for this part of the solution
u(x, y', z) = 2i 2^^ cosh kz
K III/
v(x, y, z) = 2^( 2)=? cosh kz
w{x, ?/', z) =
where a: is a pos. imag. root of sinh k]i, and ^ + pf + * 2 E 3 = .
The solutions here are obviously of the type ^ = cosh kz'~Ei 3 (x', y').
(iii) Transitory modes, 0<p or dilatational type.
Looking to (64), (82), put
E 4 = work difference from the system [ j. = cosh kzG *R
Then
, , , ,. ydE A
8^(a+ l)»c 2 7t(cosh 2kA + 1)
6 — cosh 2kA •<£
2kz' sinh kz' + (cosh 2kJi + a) cosh kz'
<(x, y, z) = ^ kE 4 { 2k«' cosh kz' + (cosh 2k/i  a) sinh kz'}
(99)
(100)
where k is a zero of sinh 2kJi + 2kJi with pos. imag. part, and ^ + pf + k 2 E 4 = .
The solutions are of the type (p = cosh kz'E 4 (o;', y'), = cosh 2k/V<£ .
II. Flexural part of the solution.
(i) Permanent mode.
Let F x = work difference from the system
3 (■ JL8 2
32ir/Ji 3 V X 6 " V X
I'a^*^**^)
Then V ,r $i = work difference from d>= — — — , z V 2 x =  ,
and ,, , » , dF, 2 / tt + 5 a+11 _d „,_
'(to 1
/ / / , N  ^F x 2 /a+5 , a+11
w(x, y', z) F, + { A/V   2 ' 2 + ~ (^  ft')}V' 2 F 3
(101)
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
195
Here V /4 F X = , and if we write (« + 1)F for F : + ±h' V f *Fi , these expressions reduce to
the form which we have taken throughout for this kind of strain, namely,
/ ' / \ d
u(x,y,z)= M
v(x\y',z')=^r
w{x\ y, z) =
where V' 4 F = .
(a + l)(z'FAz' 3 V' 2 F) + 2(^' 3 /<Y)V'F
(a + l)(Fz' 2 V' 2 F) + 2(z' 2  h 2 ) V ' 2 F
(ii) Transitory modes, ^ or rotational type.
Put F 2 = work difference from the system \^ = — ~ — sy sinh kzQqkR, .
Then
d¥_ 2
dy
dy
v(x\ y\ z) = 2 ( _ 2 ) ^J? si
«"(*', 2/', z') =
^
(  2) —  > sinh kz'
,72 F ^/F
where k is a pos. imag. root of cosh kTi , and vrf + j?f + k2 F 2 =
The solutions are of the type \^ = sinh kz'F^o/, y') .
(iii) Transitory modes, 6<p or dilatational type.
sinh kzQ () kR
Put F 3 = work difference from the system j 8 7 r / u.(a+ l) K 2 /i(cosh 2k/i  1)
I 6 =  cosh 2kIi • </>
Then , , , , x y <?F 3
u(x,y,z) =Zd — 4
K dy
2kz cosh kz' + (a  cosh 2kJi) sinh k.^'
(101')
(102)
(103)
w{x\ y, z) = 2, kF 3 J 2kz' sinh kz'  (a + cosh 2/c7i.) cosh kz }_
where k is a zero of sinh 2*h — 2kJi with pos. imag. part, and r+ rjf + k 2 F 3 = .
The solutions are of the type <t> = sinh kz' F 3 (x', t/) , = — cosh 2k/7 ■ .
34. Form of the solution for edge tractions deduced by another method.
We have thus shown that the most general deformation of a finite plate under edge
tractions only is compounded of the types specified in (98) . . . . (103). The deforma
tion is of the same form as that given by our infinite plate solutions for any part of the
solid free from body force or surface traction, and it may be of advantage to show in a
direct manner why this should be so.
Suppose, then, that we have given a displacement (u, v, w) of a finite plate bounded
by an external edge S and one or more internal edges S', the only applied forces being
tractions on the edges. Imagine the plate continued inwards and outwards so as to
196 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
form a complete infinite plate. By the general existence theorem of the subject, there
exist values of it, v, W in the space within an internal edge, continuous at the edge with
the values of the displacements of the original solid, and produced by edge tractions
alone. Similarly, if we take any surface S", within the infinite plate, but completely
enclosing the edge S, there exist values of u, v, iv continuous with the original u, v, w
at the external edge, and becoming zero on S" ; these also being produced by edge trac
tions only, namely, on S and S".
If, then, we take u, v, w to be zero outside S", we obtain altogether a system of dis
placements continuous throughout the infinite solid. The forces required to maintain
this system are given directly by the general equations of equilibrium. These forces
form areal distributions on S, S', S", and are measured by the discontinuity of stress at
these surfaces. Further, on the whole they make up an equilibrating system. But we
have shown in the preceding pages how to find a solution for such a system of force,
this solution giving displacements of order log R at most, and stresses of order Rr 2 at
most, at a great distance. Only one solution fulfilling these conditions being possible,
our solution is the solution.
Hence, finally, any displacement of a finite plate under edge tractions only is of the
same form as that given by our infinite solid solutions for a certain system of areal
force, distributed partly over the edges, and partly over an arbitrary external surface.
This is what we proposed to prove.
35. General solution for an infinite solid under any forces.
It is now easy to determine the most general form of displacement of an infinite
solid, under null body force and face traction, and free from singularity at a finite
distance. For if u, v, w be any such displacement, then within any surface S, however
distant, we have proved that u, v, iv are given by the absolutely convergent series
(98) .... (103).
If we take a right circular cylinder for the surface S, the functions F which satisfy
equations of the form j 2 + ,—, + k"F = can be expressed in series of the form
^ J m *p(A m cos Mini + B m sin wco) ,
TO
and the only restriction on the coefficients A m , B m is that they must make the double
series in which the complete solution is thus expressed absolutely convergent for all
values of p, however great.
The most general solution for any system of force applied at a finite distance is of
course obtained by adding to this complete free solution the particular solution already
investigated. It may be observed that this final result might have been obtained in
one step by the process of § 33, if in that article we had taken for u, v, w any displace
ments under given body force and surface traction, instead of under edge traction only.
The identity of the results of the two methods will be seen to depend essentially on the
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 197
fact that in the solution for a single force in any direction, the component displacement
in that direction is symmetrical in the accented and unaccented coordinates, a theorem
analogous to a wellknown property of Green's function in Potential Theory.
It is interesting to observe that, in the process suggested in the last sentence but
one, we only need to know the comparatively simple source solution for a single Z
force in order to deduce the w displacement for any system of forces and face tractions
whatever.
36. Application of Betti's Theorem to the problem of given edge tractions.
In the remaining pages, we shall be occupied almost exclusively with deformations
of a finite plate under edge tractions only. For brevity we may refer to such deforma
tions as free.
The formulse (97) express the internal displacements in terms of the edge dis
placements and edge tractions. We may indicate here the general lines along which
we naturally proceed in the attempt to reduce these formulae to expressions in terms of
displacements alone or of tractions alone.
Taking the first equation of (97), for example, if we wish a formula containing edge
displacements only, we look for free displacements in the form of functions u{, v/, w[
of x, y , z, such that u x + u{, i\ + v/, ■w 1 + w{ shall be equal to zero at the edge.
If X/, Y/, Z/ be the edge tractions in the system uj, i\ r , iv/, then by Betti's Theorem
f ((Xtti + Yvi + Z<  X>  Y x 'v  Z/wW = 0,
and by addition of this equation to (97),
u(x\ y, z') =  ff { u(X x + X/) + v(Y 1 + Y/) + w{Z l + Z/) } dS.
The problem of arbitrary edge displacements is thus reduced to a problem in which
these displacements have a comparatively simple form.
When we attempt to find a formula in terms of edge tractions only, the procedure is
not quite so simple, in consequence of the fact that the tractions X t , Y l , Z x are not equili
brating, but equivalent to a negative unit X force through (x f , y', z'). From various
methods of meeting this difficulty we select the following as the most convenient in the
present case.
We have seen in § 30 that the system i^ , v^ , w 1 can be decomposed into four systems.
The first system, say \J l , V x , W x conveys no resultant stress ; the second system conveys
a stress equivalent to a unit X force through the origin, and the displacements are
independent of x', y', z' ; the third system conveys a couple z' in the plane zOx , the
displacements contain z' as a factor, but are otherwise independent of x', if, z' ; the
fourth system conveys a couple — y' in the plane xOy , the displacements involving
x', //, z' only in the form of the factor y' .
198 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
The displacements u. 2 , v 2 , w a and u 3 , v 3 , w 3 are similarly decomposable into equilibrat
ing systems U 2 , V 2 , W 2 and U 3 , V, , W 3 with other systems conveying resultant forces
and couples. The contributions to u(x', if, z') . v(x', y f , z') , w(x', y', z') in (97) from
the various systems conveying forces and couples amount on the whole merely to a rigid
body displacement of the plate. If this be neglected, then the value of u(x', y', z') r
for instance, becomes simply the work difference from the system \J 1 , Y x , ^SVi , the
edoe tractions due to which are equilibrating, and can be balanced by a free system
U/, V/, W/. We then obtain from (97)
u(x', y\ *')=(({ X(U X + U/) + Y(V, + V x ') + Z( W 1 + W/) } ^S
and similarly for v, w.
37. Exact solutions of special problems for a circular plate.
As already stated, we are not at present in a position to complete the solution of
the problem of arbitrary edge tractions, even for the simplest form of edge. The
method just indicated may be used, however, whatever be the form of the edge, to<
obtain approximately the boundary conditions which define the permanent part of the
solution. But before entering on this important application, we shall consider a few
special problems which admit of exact solution. All of these have reference to a plate
bounded by a right circular cylinder, with or without a concentric circular aperture,
and to systems of displacement symmetrical about the axis.
The radius of the external edge is a, of the internal edge b ; and the axis of z
coincides with the axis of the cylinder, u, v, w are the displacements in the directions
in which the coordinates p, «, z increase.
Problem 1. Symmetrical transverse displacement.
The displacement v, in the most general case, is given by a series involving cosines
and sines of multiples of w. We can determine the symmetrical term of the series.
This constitutes the whole solution when the plate is subjected only to symmetrical
torsional force.
For a transverse force Q x applied at the point (p x , ^ , z x ) we have seen in article 29
that the solution is
. cU
* = j
a 1 dff
U = ~ _ 7
1 d$
n 1
47T / U,(a+ 1)
The solution for a constant linear distribution of transverse force on the circle p = Pi,
z = z x , of intensity ^ 1 /27r/o 1 per unit length, is found by integrating this with respect
to <t>i from to 2tt, and dividing by 2tt. The result of the integration is simply to
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 199
■eliminate all but the symmetrical part of \^, and to eliminate , cp altogether. The
solution is therefore,
when p> p 1}
_L £1
&TTpJl p
+ ~ — 7 2i cosh kz y cosh kzJ '/c j o 1 G 'kp , (k a pos. imag. root of sinh kK) \ 0)
 „ — ? ^j sinh K2 X sinh kzJ ' Kpfi^ Kp , (k a pos. imag. root of cosh k7i)
when p <Pi, P and ^> x have simply to be interchanged.
The only stress across a cylinder p is
Hence if Q„ , 6 be the transverse components of the traction on p = a, p = b, and v x the
transverse displacement at (p x , «j , z x ) produced by this traction, Betti's Theorem gives,
as in (97), for the permanent part of v x ,
i \\ ^ = i ( ffnA + v^p W + i f fn b ±ds b .
lir] o QirpJiJ J \ a a J 8TTfjLhJ J p 1
Also, since the distributions i1 a , Q 6 have equal moments about Oz ,
affn a dS a + bjjih > dS b = .
In the case of a symmetrical deformation, we have therefore
ii
1 ip.h\p l
\(Lb\t
Bl)1 n b dz+^ Va
a 1 J J h a
The displacement i\ = p x being merely a rigid body rotation about Oz, the permanent
solution is practically
h 2 f h a 2 f h
A ± n b dz=^—\ n n dz .... (ii>
v,=
This might have been got at once by omitting the term v — p^ 8tt /mhp from the source
solution, in accordance with the method explained at the end of the last article.
For a uniform system of couple about the edgenormal JQ b dz vanishes, and the
permanent displacement is rigorously null.
For the transitory part of the solution we will, to simplify the algebra, suppose the
cylinder solid.
Further, this not being a case where the separation into odd and even parts in z
is of much consequence, we may shorten the formulae by combining the two k series
into one.
Thus cosh k (h + z) cosh k (h + z x ) being obviously equal to cosh kz cosh kz^ when
200 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
sinh kIi = 0, but equal to — sinh kz sinh kz x when cosh *h = 0, we may transform (i) if we
put h + z = £ , & + Zy  {J , into
v =  — — + ~ — r 2i C0S h K £i cos h K CJo' K Pi ( 3'o'' <: / ' ( K a P os  i ma g root °f sinn 2k ' 1 ) • • (i")
07T/A/1 p 2trfjJl K
For
u = G Vp , we have when p = a, pw =  ^(2G 'Ka + KaG Ka) .
Hence, when the transitory part of v in (iii) is balanced, the solution becomes (p > p x )
v =  —  n +  — Zu cosh *£, cosh «£J k Pi [G k P  " ° J Kp I . (iv>
07rp/i p 27rp/? k \ UJ k« + KaJ «a /
This gives at p = a,
J +  —  £j cos h «t,i cosh k^Jo'kpj
t> =
8jr/xh a 2nfjJi K V 2 J 'ko, + k« J n /<a
Hence, for the free displacement at (p u z x ) under symmetrical transverse traction i2 a on
p = a , Betti's Theorem gives (omitting the rigid body rotation)
v x = ~2i «T 'kp, cosh at J  — — )/ cosh xtCl a d'C . . . (v)
1 (Ji « ° ri _1 V 2J ' K a + KaJ Ka) J o s " s v 7
From this
( p( ,) 1= J_X ^ p + ,(pJ T oV cosh < f cosh *OV£
The series passes continuously, as p increases to a, into the limit
— Zj cosh k£j / cosh K^f2 tt ^ , provided this latter series converges.
By Fourier's Theorem we know that it does, namely, to the value & a , it being noted
that I fi a d£ = 0. The solution is thus verified. Of course, it could easily be obtained
by the Fourier method ab initio.
The series (iv) converges very rapidly unless p and p 1 are nearly equal. By an
application of the Residue Calculus, it may be transformed into a series in which the
functions of p , pi are the fluctuating functions, and the functions of £ , £i the con
vergence factors. For consider the function of k,
1 cosh k(2/i  £) cosh k£ x , , / „ , 2G '»ca + KaG «a T , \
Vp.~ sinli 2k1i J "" ,, V lKr 2J 'Ka + KaJ oK a ° * P ) '
It is easy to see that log * disappears from the last factor, and that the whole function
is a uniform, odd function of k; also that if ^>^i,jo>j°i, the function vanishes at
infinity in such a way as to make the total sum of its residues equal to zero.
The poles of the function are k — 0, the (pure imaginary) zeroes of sinh 2ich, and
the (real) zeroes of 2J '/ca + koJ^cci. The function being odd, we have (series of residues
at pos. imag. roots of sinh 2kJl) + (series of residues at pos. roots of 2J / /c« + K«J /ca)
+  residue at (k = 0), equal to zero.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 201
The first series of residues is the function v of (iv). Thus we obtain, (£> <^)
4 "V cosh K(2h  £) cosh k£j 1 T , T , ^
y ~^« sinh 2kK ^ a 3 ( J oKa) 2 J o«P 1 Jo^ I
When ^>^,we have merely to interchange £ and (^ in this formula.
We may verify in a moment from this, as from the perfectly equivalent form (iv),
that the internal couple is balanced by the stress at the cylindrical boundary, and that
there is no stress across the plane ends. But if we remove the last three terms from
(vi), we make no change in the internal singularity, these terms being the same whether
£ or £ x be the greater.
We thus obtain the displacement when the internal couple is balanced at the plane
ends, namely, (£>£i)
2 y , 4 "V cosh k(21i  1) cosh kL 1 T , T , , ■•,
TTfia* ^ s Tr/jLCt k sinh 2 K h K 3 a 3 (J Ka) 2 ° Hl ° ' v
Here, as in (vi), the summation extends over the positive roots of 2 J '>ca + /caJ /ca . The
solution for symmetric transverse traction & 2h , Q Q on the ends, which might be obtained
in an abnormal form from (iv) with the cognate formula for p > p 1 , is given in normal
form by a direct application of Betti's Theorem to (vii).
Thus
v (Pl . «  i 2 cosh ffifa — i r w ,v P
/xa K sinh 2kA K 3 a 3 (J /<«) 2 Jo
, 8 y coahK(2hL)J' KPl 1 /■» T , ,
+ — ^ ■ , o' ; XSTt T2 V pJ K P d P \ ■ • ( vm >
/xa * sinh 2k/j K J a 3 (J Ka) 2 Jo
4 /""
" 7—4 Pi^i "oP 2 ^P
The result belongs rather to the theory of a long rod than to that of a thin plate. The
permanent term depends only on the integral couple, and coincides with that given by
Saint Venant's theory of Torsion.*
38. Problem 2. Boundary values of the normal displacement u, and the shearing
stress normal to the plate ^>, are given functions symmetrical about the axis ;
the displacement v, or the shearing stress ^, vanishes.
We begin with the case of a solid cylinder.
(i) Permanent extensional mode.
Referring to § 33, I. (i), we see that under the conditions proposed the function
Ei must vanish, and the solution in cylindrics is
. ap 1 a + 1 ap l
w (fti»»i) B3 ^Xi 2 iVi 2E 2
a+ 1
* The writer hopes to publish shortly a solution of the problem of equilibrium of an infinite circular cylinder,
in which the celebrated solutions of Saint Venant will appear as the leading terms. It will be shown that in a
fiuite cylinder the permanent modes are given exactly by Saint Venant's theory. In the theory of thin plates, the
permanent modes can only, in general, be found approximately.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 31
202 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
where E 2 =    . work diffce. from 8 = cj) = x(p) — z 2 log p + \px log p,
or E 2 = — ^ —  . work diffce. from 6 = cp = log p , the part omitted being merely
a constant.
Now in the system 6 = <p = log p
«(o+l)/p
^=2/t(a+l)/p»
w =
£=0
Hence the solution
M=  (a+ l)p/a 2
p >2Ma7)/a*
w= 2(a3)z/a 2
^ =
cfc .
will, taken along with 6 = (p = log p , give it and p~z = at /> = a . The balanced solution
gives p~p = — 1 6m/« 2 , w = 2(3 — a)2/a 2 , at the edge.
Hence
and
w(p 1 ,z 1 ) = 2 — gj 8a7i7 a I 2/t J
(ii) Transitory extensional modes.
The solution is given by (100) with
E ^ 8^ + lATcosh2K/ t + l)  W ° rk dlffCe  fr ° m the S ^ Stem {fr^S^
In the system mentioned
M = /<G 'Ko{(cosh 2/<fe + a) cosh kz + 2kz sinh kz}
^ = K G Kp{(cosh 2i<ha) sinh «2 + 2k2 cosh kz}
__ IE = G 'Kp{(cosh 2kJi+ 1) sinh kz + 2kz cosh kz}
— £^=  G Kp{(cosh 2k7i + 3) cosh kz + 2kz sinh kz}
 — G 'K/o{(cosh '2kJi + a) cosh kz + 2kz sinh kz} •
The balancing system for u and z^ at p = a is therefore
tf>= — — ^ — J Kp cosh kz , 6 = cosh 2k/i<£ .
Iii the balanced system, at the edge
, — < (cosh 2k1i  a)sinh kz + 2kz cosh kz >
r kcl \ J
.££ = — — ^ — < (cosh 2 kIi + 3)cosh kz + 2kz sinh kz >
2u aJ n kos I v J
to
ot J I
Hence for the free solution with edge values u = u a , T P = Z a ,
9 ° (cosh 2/c/i  a sinh kz +
T ^ash 2^ + 3 cosh*
2kz cosh kz) I ,
kz + 2kz sinh kz
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 203
(iii) Permanent flexural mode.
The solution is given by (101) with
3 1
F, = 5 . — p, 2 • work diffce. from 6 = —<f> = z log p .
32rrp.h 3 4
^ =
^ = 2/x(o+l)z/p 2
0= <£ = zlogp, W=(a+l)z/p
In the system w = ( a + l)log P
Hence the solution balancing u and T P at /o = a is
M=(a+l)z/o/a 2
to =  (a + l)p 2 /2a 2 + (a  3)z 2 /ffl 2
« P =
p P =2fjL(7 a)z/a 2
In the balanced solution, at the edge
to = (a + 1 )(log a  I) + (a  3)z 2 /a 2 ; pp = 1 6p.z/a 2
The constant term in the value of it' will disappear since I zpdz = § . Thus
F >  34= „•/*_{!< V  ** }*
and
u (pi » z i) =  z iPi
/ \ 1 8,3q 2 > ■ ■ a J ^(a3)z 2 8zw a V
^ a+l '
cfe
(iv) Transitory Jlexural modes.
The solution is given by (103) with
F * = 8^(a + l)^(c OS h2^l ) ' WOTk diffCe " fr ° m the SyStem { t= G °Zsh2 K h. 4>
In this system
u = kG 'i<p{ (a — cosh 2 Kh)smh kz + 2 kz cosh kz}
w = kG kp{  (a + cosh 2k1i) cosh kz+ 2kz sinh kz}
~2 k^ — (V*/°{(1 ~~ cosn 2kA) cosh kz + 2kz sinh kz]
~v o~~ ~ O Kp{(cosh 2kIi  3) sinh kz — 2kz cosh kz}
—  G '/cp{(a — cosh 2k1i) sinh kz + 2«z cosh kz)
k P
The system balancing u and zp at p = a is
4> =  Yn — J Kp sinh kz , 6=  cosh 2kA • <p .
In the balanced system, at the edge
w — , , <  (a + cosh 2k1i) cosh kz + 2kz sinh kz >
ao ku * )
^ = — ^ — < (cosh 2/cA  3) sinh kz  2kz cosh kz >
2u ao n ko, [ )
sp. ao i
Hence for the free solution with edge values u = u a , zp = Z a ,
J g^(  a + cosh 2kU cosh kz + 2kz sinh kz) \ _,
h \ + kuJS  cosh 2kJi sinh kz + 2kz cosh kz) )
If the given values of Z a , u a are the same as the edge values of z^ , u , in one of the
particular solutions, then clearly this particular solution by itself is the solution, and the
Y = I J 0*Pl C h { ^a
3 2(a+ l)/c 2 /i(cosh 2kK 1) J 'ica ^ 2^
204 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
integrals which define the coefficients of the other particular solutions must vanish,
while the integral corresponding to the solution left has its value determined. These
results are easily verified by actual integration.
This remark may be used to find the solution for a hollow cylinder, which of course
might also be obtained directly by the above process. We shall illustrate the method
by finding the value of F 3 corresponding to any given root k of sinh 2k]i — 2kJi , when
we have given u a , u h , Z a , Z 6 .
This value of F 3 we know is of the form AJ Kp 1 + 'BG Q Kp l .
The complete values of u and of ^/2m for p = p x are given by series which manifestlv
converge uniformly so long as & < ^ < a .
Multiply the series for zp~/2ju. by  (a + cosh 2k//) cosh kz + 2kz sinh kz ,
the series for u by k(3  cosh 2*7t sinh kz + 2kz cosh kz) ,
add, and integrate with respect to z from — h to h . All the terms disappear except
that associated with the given root k. We thus find
(AJ ' KPl + BG '/cp 1 )2(a + l)/c 2 7i(cosh 2 K h  1)
— / )9~(p = p 1 )(a + cosh 2kJi cosh kz + 2kz sinh kz) + ku(p = p x ) (3  cosh 2kJi sinh kz + 2kz cosh kz) > dz .
This is proved for the case b<p 1 <.a . Now take the limits of both sides for p 1 = a .
The limit of the integral is found simply by replacing Tp and u (p = p^ in the integrand
by zp and u (p = a) , provided the resulting integral has a meaning, which will be the
case if zp and u (p = a) are integrable functions of z. Similarly we may take the limit
for p x = b , and thus obtain two equations to determine A and B.
It will be observed that by this method we avoid two difficulties which in problems
of this kind are often introduced unnecessarily by physical writers, namely, (i) the
difficulty as to the convergence of the series for 7 P and u, when the value p x = a or
b is substituted term by term, and (ii) the allied difficulty as to the continuity of the
series right up to p = a or b, even when it is known to converge. Judging from
analogy, we may feel reasonably certain that the series will in fact converge at the
limits, at least in the case of ordinary functions ; but it is worth while noting that,
whether they converge or not, the Fourier method of assuming the continuity and
convergence, and determining the coefficients by integration, does give the correct values
of these coefficients.
On the other hand, while our ' Green's function ' method proves definitely that
any possible solution has the form given above, it does not prove that a solution is
possible for arbitrary edge values of Tp and u. The investigation might be completed
by verifying that the solution obtained does actually satisfy the conditions, which
would not be difficult in the present case. Alternatively, we may rely upon physical
considerations, or upon a general analytical existence theorem. The proofs of theorems
of this type in other branches of physical mathematics have been considerably improved
within recent years by Pom care and others, and their methods are equally applicable
to the elastic equations.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 205
39. Problem 3. To determine the permanent modes, having given the
symmetrical edge tractions z^> and p~p.
Supposing the cylinder solid, we have only to make a slight modification in the
process of (i), (iii) in last article.
Extensional mode.
In (i) the balancing solution must now be taken as
_ ( a+1) 2 _p 2(3a)(tt+l) £
U ~ 7a a 2 ' W ~ ~ 7a a 2
This, along with = cp = log p , gives p~p and p~z = at p = a .
The balanced solution gives at the edge
8 a+1 2(3 a) (a+1)
« =  = , w— — s
a I a ' 7a
z
Hence
M (n, . Z. ^ = n. 1
i+l 1
M (pi » z l) = Pi j
* ' " ' ' a+1 1 /•"
. „ a  3 } ~ firjr (4«P„ + a  3 zZ a )<fe .
(/°i. z i)= 2 ^n z ij ^«i6m;^ v
This gives the permanent mode exactly. The ordinary approximate theory omits the
term in Z a from the integral.
In Chree's solution of the problem of a rotating disc, the stress zp vanishes identi
cally at the edge, while I F a dz = . His solution is therefore exact, so far as the
fundamental mode is concerned.
As in last article, we infer from the form of the above solution that for any other
than the permanent mode / (i P ^ + a 3 z ? p ) = .
This may be verified by actual integration. Farther, in the case of a hollow
cylinder the solution is of the form
m (pi>*i) = M + b /pi )
«( Pl! Z 1 ) = 2A(a3)z 1 /(a+l)}
and the coefficients A , B are found from the conditions that
I (4p • p P + a  3 z • 7p) dz must, for p = a and p = b
have the same value for the assumed form and for the given tractions.
Flexural mode.
Referring to (iii) of last article, the solution balancing
6 = cp — z log p at p = a is now
„__(<*+l) 2 ZP (a+1) 2 P 2 (3a)(a+l)z 2
W^ +
7a a 2 ' 7a 2a 2 " 1 " 7a
The balanced solution gives at the edge
o a+ 1 z (3a)(a+l) Z 2
u ~ _0 7 — ; w = const + = »
( — a a I — a a
206 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Hence
«( Pl ,.:,)= z lPl ,
The ordinary approximate theory takes account of the term in zP only.
The solution for a hollow cylinder can be obtained by this method, or by taking, in
the notation of (94),
F = A P {~ + B log Pl + C (i Pl 2 log Pl  W) ,
determining C from the value of / TpAz at either edge, and then determining
A and B from the conditions that
I (  8 P z pp + 3  a z'zp) dz must for P = a or P = b
have the same value in the assumed form as in the actual displacement.
40. Expansions of arbitrary functions.
When we attempt to apply the method of last article to the determination of the
modes corresponding to the various roots of sinh 2/e/i ± 2ich, we are at once confronted
with an apparently insuperable difficulty. The determination of any one mode is
reduced by the application of Betti's Theorem to the special problem of balancing the
particular source solution involving a given root k. Now in similar investigations
connected with Laplace's equation, the equation of conduction of heat, and other partial
differential equations of the second order which occur in physical mathematics, the
analogous balancing problem can be solved without difficulty for certain simple forms of
edge, and the balancing solution is of the same type as the particular source solution,
that is, involves only the same root *. In the present problem, however, the balancing
solution will in general involve particular solutions of all types, as will be seen below.
Various theorems relating to the expansion of arbitrary functions may be found,
similar to the theorem suggested at the end of § 38, but these do not help us, at all
events immediately, to the general solution sought. One way of obtaining these
expansion theorems may be indicated here ; the method is of very wide application.
On a circular cylinder p — a, within the infinite plate, let areal force be distributed,
the components of its intensity per unit area being P cos m« , Q sin m« , Z cos m<o , where
P, Q, Z are functions of z.
The infinite plate solution for this distribution of force can be written down, and
the components of stress ^, ^ , ^ , calculated. These are given in different analytical
forms according as p is greater or less than a. The expansion theorems are derived
from the conditions of equilibrium
with two similar equations for , Z.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 207
There are other expansions which we know must exist, but the coefficients of which
we cannot determine. The following examples are of special importance, and will be
found useful immediately. They refer to the case of a plate bounded by a single
infinite edge, filling, say, the region a?>0, and the displacements considered are such
that u, w are functions of x, z only, while v vanishes.
Extensional modes.
The permanent mode is of the form u = x, w= — ' z .
In a transitory mode with (p = e iKX cosh kz,6 = cosh 2>ch ■ <f> , we have
o — — 2 = ie lKX (cosh. 2k1i + 3 cosh k.c + 2kz sinh kz)
o ,• 2 = e' Ka; (cosh 2kJi + 1 sinh kz + 2kz cosh kz)
As a special case of the results of last article, it follows that the coefficient of the
permanent mode is determined from the given value of I xx dz at the edge ; and
this integral normal stress is zero for each of the transitory modes.
Hence if P(z) be any even function of z, with J ~P(z)dz = 0, and Z(z) be any
odd function of z, coefficients C K exist such that at the same time
U
2
= q 41 C K j'e i,c:r (cosh 2/c/i + 3 cosh kz + 2kz sinh kz) = P(z)
an
U
(i)
x = o Zj C K e !,c:,: (cosh 2k?i + 1 sinh kz + Ikz cosh kz) = Z(z)
Flexural modes.
In the permanent mode F of equation (95) is of the form Aa^ + Ba: 3 . A and
B are found from the edge values of f^z.Pxdz and T Zdz. These integrals
vanish for a transitory mode, in which, with <f> = e iieX sinh kz,0=  cosh 2/ch ■ <p,
XX
2/xiK
xz
2~r~ K i = ie lKX (3  cosh 2 K h sinh kz + 2kz cosh kz)
—  2 = e lKX (l  cosh 2k/i cosh kz + 2kz sinh kz).
We infer that if P(z) be any odd function of z, with f zP(z)dz = 0, and Z(z) any
even function of z, with j h _Z{z)dz = 0, values of C K exist such that simultaneously
x = q^Li C K ie iKX (S  cosh 2k1i sinh kz + 2 K z cosh kz) = P(z)
L^
= Zj C^^l  cosh 2k1i cosh kz + 2*z sinh kz) = Z(z)
a; =
(")
The limit for x = may be taken term by term, provided the resulting
converge. In the following analysis we shall assume that they do so, but this is
merely in order to avoid lengthy forms of statement ; the argument could be put, if
necessary, in a form independent of this assumption.
208 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
41. The problem of given edge tractions for a thin plate.
The form of the complete solution is exactly known, and the three boundary
conditions in their exact forms could, therefore, at once be written down. The whole
strain is compounded of an infinite number of modes of equilibrium of known types,
and it is obviously suggested as the method of attack that we should try to disentangle
from the general boundary conditions those special conditions by which each mode is
separately defined. When the plate is thin we find that within certain limits this
can be done, and, in particular, the conditions defining the permanent modes, which
in the case supposed are incomparably the most important, can be found with con
siderable exactness.
We shall understand that the edge traction, or any component of the edge traction,
is given as a function of x , y , z/h or of s , z/h , where s is the arc of the edge line, so
that if £ be put for z/h the form of this function is completely independent of h. The
theory may be applied to cases in which the proviso is not fulfilled, but before such
application the given traction is to be separated into parts of ascending order in h , say,
for example, f(x ,y ,'Q + h f(x ,y ,Q + h 2 f 2 (x , y , £) f etc. ; then for a first approxi
mation we deal only with f(x , y , £). The theory does not contemplate such a
distribution of traction, as, for example, sin (ms/h) , m being a number, where the rate
at which the traction varies along the arc is of a lower order in h than the traction
itself.
The trace of the cylindrical edge on the middle plane of the plate is the edge line ;
the outward normal, and the tangent, to the edge line will be referred to as the
normal, and tangent simply ; the generator of the cylindrical edge at right angles to
these at their point of intersection may be called the perpendicular.
Let I, m be the direction cosines of the normal,
then —m, I are those of the tangent.
The normal displacement is p = lu + mv
and the tangential displacement a = — mu + Iv .
The tractions on the edge in the directions of normal, tangent, and perpendicular,
are nn , ns , nz or N , b , Z.
42. Extensional strain.
In this case N, S are even functions, and Z an odd function of z.
It will be advantageous to express as far as possible the displacements and tractions
at an edge in the various types of solution in terms of derivatives along the tangent
and normal.
Alongside the symbol a we shall use the more familiar <r, the relation between the
two being given by « + 1 = 4(1 — o) ; 3 — a = 4cr.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
(i) Permanent mode.
U  {j+ \cr2 dx\dx dil)
, 7 , a z" d / dU , dV
102 dy\ dx dij
'dU dV\
— _  M
where
209
lo \ da; d ;/ /
Put
2 ±(dU + dY\ + n_ v \±(dU_JV\ B0 )
dx\dx dy) dy\dy dx) \
2—(— + — } (lo) — (— — ) = I
dy\d.x dyj (ix\dy dx)
A _ 1 /dU dV\ _ 1 fdu dv
1  (T\dx dy / 1  o \dz dv/
n= j(*?_rfv
\dy dx
Then
da; dy
dA _ dil _ Q
d?/ da;
 and at an edge
du dv
dy dx
dn ds
dA_dII
ds dn
=
Also
1 — cr Vda: dy
. 2u / du dv\
yy= \^VTx + ^y)
~ 2u lo /dw dv
wy = ^ • — — I — !
lo 2 \dy dx/
The components of traction parallel to the axes are
X = Ixx + lllxij ; Y = lxy + TWyy ,
These are easily transformed into
Hence
also
or
X = 2//zA +mn s j; Y= 2/*(  /H + m A +
m = ZX + n»Y = 2uf A + m—  Z — V
\ ds ds)
r s = mX + /Y = 2a( U + l^+mtY
ds
ds)
,= 2/ /a s.£.y
\ as p )
s = 2 "( n+ f)J
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8).
32
210
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
If P. Q be the values of p, q when z = , then
r, , <,dH /w , 1 'yd A
cis as
2yu. ds p 2 \ ds' p ds J
2fJL
„ dP Q 1 2 /d 2 n ,liA
=  IT + —   2. — —(TZ\ tr^ +
ris
£?S 2
rfs
(ii) Rotational Transitory Modes.
dy
v=2 ( '±\
dx
2/a cfa rfy
5. = ^i.  ^t
2/x dy' 2 dx'
2/x, dy dz
Taking the axes of x and y for a moment along the normal and tangent, these
give at once by means of (k)
n d\lS
q=2 d ±
dn
nn
nd d\j/
ds dn
2dxjj
p ds
\
ns
27"
dz 1 p
# + 2
dn
d>b
nz
. <*ty
2p.~
dsdz
.
The function \J/ can be expressed as a series of terms of the form
^„(a;, y) cos? ^ , where V fy»  ^r<A,< = ° •
Hence in cases where the values of \^ along an edge are given independently of h,
or generally, when the rate of variation of 4" along an edge is of the same order
in h as ^ itself, say order zero, terms of various orders occur in the expressions
for the displacements and tractions.
Thus
dj/ dhf,
ds ' «V
are of order
dif/ d d\\i d d\j/ ,
dn ' ds dn ' ds dz
d\f,
dz 1
2
It follows that in such cases this type of strain contributes mainly to the tangential
displacement and traction at an edge.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
211
We see also that the principal part of the displacement is of one order higher
in h than the principal part of the traction.
(iii) Dilatational transitory modes.
There would be some advantage in working with the functional symbols 6, (p,
as with ^ in the last case, but on the whole it seems clearer to deal with a typical
solution corresponding to a single root k of sinh 2<h + 2kJi.
<£ = cosh kz/(x, y) ; = cosh 2k1i • <£
dx
dy
(cosh 2/c/i + a) cosh kz + 2kz sinh kz
to = k/ { (cosh 1 Zk1i  a) sinh kz + 2kz cosh kz }
~=  K*f \ (cosh 2k1l + 3) cosh kz + 2kz sinh kz I
— —hi \ (cosh 2kJi + a) cosh kz + 2kz sinh kz \
dy 2 I v J
—  = — —  — \ (cosh 2k1i + a) cosh kz + 2kz sinh kz \
2/x dx dy V J
~ = K^f 1(1+ cosh 2k/;,) sinh kz + 2kz cosh kz \
2u dx I J
Hence
nn
=  k 2 / \ (cosh 2k/i + 3 ) cosh kz + 2kz sinh kz [
 ( f + ~ ) \ (cosh 2kJi + a) cosh kz + 2kz sinh kz I
\ p dn ds'J { J
k~= \^r ^~~ — r ) 1 (cosh 2k7& + a) cosh kz + 2kz sinh i
2/x Vrts ara p $s / I
<*//
hKz\
7T = f^ 1(1+ cosh 2«/i) sinh kz + 2kz cosh kz
2/x r/«. (. J
Thus at an edge where the rate of variation of / is of the same order in h as /
itself, say order zero, the normal and" "perpendicular displacements are of order 1,
while the tangential displacement is of an order one higher ; the normal and
perpendicular tractions are of order 2, the tangential traction being of order 1,
or again one higher.
Hence this type of strain contributes most to those components of displacement
and traction to which the ^ type contributes least, at an edge.
(iv) It is now possible to assign approximately to each of the three types of strain
the portion which it carries of any given distribution of edge traction. Let this
distribution be N, S, Z, functions of z, s, of order zero in h. We can satisfy the con
ditions to the first order by a solution in which the principal part of the traction
212
MK JOHN DOUG ALL ON AN ANALYTICAL THEORY OF
is of order zero for each type of strain. For, taking account only of these principal
parts, the equations to be satisfied at the edge are on this supposition
N = N„ +N d ,
s = s p +s,
z= z d
the suffixes referring to the permanent, rotational, and dilatational types respectively.
Now
Also
S r = 2^ and j ^ S r <fe = 2/z
~dz
= 0.
N d =  2ix2t k 2 / k < (cosh 2kJi + 3) cosh kz + 2kz sinh kz) >
and, as in art. 40,
Hence the above equations give
j" A N„<fc = o.
N„ =
and these conditions determine the permanent mode,
•v// can now be found from the boundary condition
For, taking
the condition is
2Al ft = S^f Sdz.
az l 2hJ n
«A« (*, y) «os — ,
2 fc, • tf cos ?  Z =  ±j \ S  g^r/ Jl* }
Now the righthand member here is a function of s, z, even in z, the zintegral
of which from — h to h is zero for all values of s. It can therefore be expanded
by Fourier's Theorem in the form, valid from z = —h to z = h,
> . nKZ
*— > A,, cos 7—
»=1 ft
ra'V 2
#
^ n is then determined as satisfying vfy* + jr^n = throughout the plate, and taking
the value A„/w' 2 at the boundary.
Lastly, the equations to determine the dilatational mode are (since at the edge
  i K f to the first order),
* ft ' K ~ j ( C0SU 2k ^ + 3 ) C0Sn KZ + 2kZ sin ^ KZ f = _ 9~  ^ ' ~ 9T / ^^ f
./« ' **M (1 + cos h 2k/i) sinh *z + 2/cz cosh kz  =  h) Z
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 213
By the same method as in the case of \^, but using the theorem (i) of § 40,
instead of Fourier's Theorem, we see that functions f K (x, y) exist, solutions of
(y 2 + /c 2 )/*=0 and satisfying the above boundary equations.
Thus the apportionment proposed for the edge tractions does actually satisfy the
conditions to a first approximation. The solution found gives tractions of which
the principal parts are the tractions actually assigned in the problem. The residual
traction given by the solution is of the first and higher orders ; and a second
approximation to the problem will be obtained by subtracting a solution giving the
residual tractions of the first order, such solution being found by the method used
in the first approximation.
This process would be tedious, and the way would be blocked at an early stage
by our ignorance of the coefficients of the expansion (i) of § 40.
We therefore pass at once to the consideration of the powerful method furnished
bv Betti's Theorem for the determination of the permanent mode.
43. Extensional strain. The Green's Function method for the permanent mode.
The method has already been explained (§ 36). If we wish the permanent displace
ment at (x\ y', z'\ in any direction (say the displacement u), we take the permanent
part of the solution for a unit force in that direction (a unit X force), modify it by
removing the terms which convey resultant stress, and then try to balance it at the
edge by adding a solution, without internal singularity, which shall neutralise its edge
tractions.
The displacements at the edge in the balanced solution, i.e. in the solution obtained
as the sum of the source and balancing solutions, being u', v', w', or p', q', iv', and the
given tractions X , Y, Z , or N , S , Z , we have
c, y, z') = J f(Xu + Yv + Zw')ds dz
the integral being taken over the cylindrical boundary.
The thickness 2h being supposed infinitesimal, the object of the method is to deter
mine a few of the terms of p', q', w' of lowest order in h .
An alternative method would be to determine the functions E 2 , B 2 of § 33, I. (i),
in terms of edge tractions ; but the least confusing method of all is perhaps to
determine
v ,^dE, + r/E^ and y , = _<(K l + ,/K 2
dy' dx an " dx dx'
These do not contain z', but when they are known the complete solution can
obviously be written down. We begin with U', and in fact it will not be necessary to
determine V separately.
214 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
Now U' is the work difference from the system
From this system let those terms be removed which transmit the resultant stress
(equivalent to a unit X force through x' ', y', z') . Then the remaining displacements
have still the forms discussed in § 42 (i), and we shall use for them, and for the various
quantities related to them, the notation given there, modified by the addition of a
suffix in each case.
The problem is now to balance the edge tractions due to the system u , v , w .
The principal parts of these tractions, which in this case are simply the terms in
dependent of z , are balanced by a solution of the permanent type (which we shall
distinguish by the suffix 1) such that at the edge
(*»  1°4") ♦ (*.  %*)
(».♦$*) + (*+**
<i>
p J \ x ds p
These conditions define the solution with suffix 1 .
The residual tractions from the compound solution u + u t , v + v x , w + w x contain
the factor z % and are of order h? as compared with those already balanced. To balance
these residual tractions, solutions of all types are required, but, as in § 42 (iv), the per
manent solution (which will be marked with suffix 2) is determined from the integral
residual tractions ; the permanent displacements are of order h, while those from the
transitory solutions are of order h 2 , those from the source solution being of order hr 1 .
The displacements of the balanced solution are therefore to terms of order h inclusive
in the notation of § 42 (i),
(2)
and with these values of _//, q', w'
U'(x',y',z) = jf(Np' + Sq+Zw')dsdz . . . (3)
All the steps of the above process can actually be carried out in the case of a circular
plate, and the final formula gives a perfectly definite solution provided merely that
N , S , Z are functions integrable over the edge. It should be specially noted that, in
this form of the solution, discontinuity of the applied traction gives rise to no difficulty
whatever.
On the other hand, the formula does not give a ready answer to such an important
question as " What are the relations between the tractions actually applied, and the
the error in each case
being of order li?
THE EQUILIBRIUM OE AN ISOTROPIC ELASTIC PLATE. 215
tractions required to maintain the permanent solution alone ? " or the practically
equivalent question, "What conditions must the edge stress satisfy in order that the
permanent mode may be absent from the resulting strain ? "
The expression found for U' may be transformed so as 10 supply answers to these
questions.
In the first place, we note that the values of p', q', iv' will still be correct to the
order stated (but will contain superfluous terms) if in the expressions just given for
them we write
n + Hj + n 2 for !!„ + II,
and
A + \ + A 2 for A () + Aj .
Write also
n for n + iii + n 2 , A ff > r \ + \ + X,
and similarly for the other quantities.
Hence
rv = — erzA
and
U'(x' , y' , z') = f f  N(P  W^) + S (Q + Joa^)  Z^A j ds dz . . (4)
Also from (l) A  ^   ,  n + —  " are of order h , and the formula for IT' will there
' as p as p
fore be correct to the same order as before if we substitute ^ +  for A , and
ds p
j for II. We shall also write
as p
f Nr7 2 = N n , f" 2 2 Ncfe = N
.' h J h
]_ Sdz=S , f z 2 Sdz = S,
f\zdz = Z 1 .
Thus
w>j* {****&£ >**+*«&%+*)<% +!)} • w
7/N 2 , S 2 ,Z 1S ^rr, ^ 2 are continuous over each edge line, integration by parts transforms
this into
Hp(v^i.«.^f)^(M.frlH!.5)} . . (6)
216 ME JOHN DOUGALL ON AN ANALYTICAL THEORY OF
ITcnce, in order that the permanent mode should be absent from the strain due to
N, 8, Z the following two conditions are sufficient : —
W  * Z hr —* A fr  S ' 2 
ft as f> as
o '/Zj . <r ^N„ . rf'S,
b + tr  l i ^ + Ao — v 1 = U
as o as as 
(7)
at every point of the edge line or lines.
Further, all systems of traction for which the lefthand members of (7) have given
values at every point of an edge will produce the same permanent mode. Now as one
such system of traction we may take the traction due to, or producing, the permanent
mode by itself, as given in § 42 (i). This gives at once the boundary conditions satisfied
by the functions U, V of that section, and these boundary conditions, with the internal
equations
dA cm _ Q . dA _ <m =
dx dy ' rt.c dy
completely define U, V which are thus determined, to a third approximation in general.
The defining differential equations and surface conditions being practically of the same
form as in the familiar first approximation, we need not detail the proof that U, V are
actually determinate from the conditions, but pass at once to the important conclusion,
an immediate consequence of this determinateness, that the permanent strain will not
be absent unless (7) are satisfied, or, in other words, that these conditions are necessary ;
as well as sufficient. From this again it follows that these conditions are fulfilled, to
the order stated, by each of the transitory modes ; and this remark is valuable, because,
once it has been verified by direct integration, it obviously leads, by an extension of
the process of § 42 (iv), to a completely independent method of dealing with the whole
problem. The method is noticeable for its simplicity and directness, but a somewhat
serious defect is the difficulty of adapting it to the case when the edge stress is
discontinuous.
This leads us to consider the correction that must be applied to the integral (6)
when the conditions of continuity stated in connection with it are not fulfilled. It will
be sufficient to take a case in which breach of continuity occurs at only one point E of
the edge line.
We have defined the positive direction of an edge line in (k) ; let the excess of the
value of f(s) just on the negative side of E over its value just on the positive side be
denoted by [/(•?)]• Then if (p(s) be continuous
J/(S) I $( S )dS = *(*)[/(8)]  j #0 £/(8)d8 ,
the integrals, we may suppose, being taken round the edge line from the positive to the
negative side of E, and the value of <p(s) in the integrated term being taken as at E.
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
217
Then to (6) we have in general to add
Hf7) [NJ+ K§ + 7) [S * ] ' Q[ZJ 1
or, otherwise arranged,
p { if p« +, 4^]} + li { * [ZJ + J 7 [N J " K * 1 }
fWJ + SMW
The various terms of this expression may be interpreted with the help of the concep
tions of sources and doublets. Thus, to go back to (3), we see that the part of U'
arising from an element N e&? of normal traction at E has (P + P^NoC&j for principal
term. P + P x is therefore (principal term of the) value of U' due to a unit element of
normal force at E. (Since this unit element can only exist in any actual deformation as
part of an equilibrating system, the phrase due to in the last sentence must be taken
under reservation. The solution of which Po + Pj^ is the xdisplacement at (x f , y', 0)
is in fact maintained by a unit element of normal traction at E, acting along with a
continuous system of force in equilibrium with this element, and distributed over the
edge in a manner depending only on the statical value of the element, and not at all on
the position of E. For any equilibrating combination of elements, the aggregate of
these continuous systems will disappear.)
Now the first of the above integrated terms is equivalent to (p +p )£— [S.,].
Hence the discontinuity in S 2 at E has the same effect at a distance from the edge
as would have an element of normal traction distributed over the perpendicular at E so
as to give a resultant h— [S 2 ]
Again an element — A of normal traction at E, combined with an element A at E',.
where EE':
ds, will give
U' =
■AfP. + PJ*
ds
(P + P x ), if we take Ads = 1 .
— ( p o + p i) i s therefore due to a unit doublet of normal force at E, and from the term
us
o[N 2 ] we conclude that the discontinuity in N 2 at E has the same interior effect as
a doublet of normal force at E of strength — ^o[N.J.
The other terms may be interpreted similarly. It does not seem possible to account
on physical grounds for any except the principal terms of the solution given above.
The principal terms are of course the same as those deduced in the ordinary theory from
the ' Principle of the elastic equivalence of statically equipollent systems of load.'
TRANS. ROY. SOC. EUIN., VOL. XLI. PART I. (NO. 8) 33
218
MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
With reference to the equivalence of mere discontinuities to line elements and
doublets one or two remarks may be made. Discontinuity in the applied force will not
produce infinite displacement at a line where it takes place, but a line element of load,
and, a fortiori, a doublet, will do so. The permanent mode may therefore contain
infinities at the edge which do not exist in the exact solution. There is really no
difficulty in this, since the permanent mode does not purport to represent the strain,
even approximately, in the immediate vicinity of the edge. The point may be
illustrated by the permanent part of the infinite solid solution for a single force. This
becomes infinite on the perpendicular through the source in a totally different way from
the exact solution. A good deal of discussion took place at one time over a similar
point in the fiexural solution. This will be referred to again, but the considerations we
Jiave adduced seem to remove the chief part of the difficulty.
44. Fiexural strain.
In this case N, S are odd functions, and Z an even function of z.
(i) Permanent mode.
This mode is defined in terms of one function F of (x , y) satisfying V 4 F = , and
may be referred to simply as an F strain.
tf,' =  (zY  ^ 3 v 2 F) ; 6 = zY  £z 3 v 2 F  2hh^Y
dn
? =
to =
ds
4(1  <r)(zY  £z 3 v 2 F) + 2(l;: 3  /i%)v 2 F
4(1  (r)(F  kVF) + 2(?: 2  /i 2 )vl'
For shortness in writing out the stresses, we shall work with symbols ^ , &, , 9 3
denoting operations of differentiation applied to F, and defined by the equations
Then
, 1 = 4v 2 + 4(.l)(lii +  2
S \= 4(1  cr)( j ~  1 f
\as dn p ds
S., = 4 f v 2
dn
ii.n
/x2 2 v ' s
1 J_
2"1
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
2.19
(ii) Rotational transitory modes.
These are as in § 42 (ii), except that now
where
V , / x • 2n + 1 vz
n=0 z n
(iii) Dilatational transitory modes.
<f> = sinh kz f(x ,y); =  cosh 2k/i • <£
As in § 42 (iii),
(a  cosh 2k/i) sinh kz + 2kz cosh kz
I
df
* ds
w = k/ {  (a t cosh 2kJi) cosh kz + 2kz sinh kz}
25 =  «y{ (3 cosh 2k1i) sinh KZ + 2K2 cosh kz}
— ^ \ \ ) < (a  cosh 2k1i) sinli kz + 2kz cosh kz >
p ^ra as / ! J
= ( — — — _£ ) J ( a — cosh 2kJi) sinh kz + 2kz cosh kz >
\ds dn p ds/ { )
, (1  cosh 2k1i) cosh kz + 2kz sinh kz }
2fj. dn\ x ' f
nz _ df
We have
f =  i K f to the lowest order,
dn
or /=
i df
k dn
Hence if we put
the above strain gives
df
dn y *
i{(cosh 2k/i  3) sinh kz — 2kz cosh kz) = N(kz)
(1  cosh 2k7i) cosh kz + 2kz sinh kz = Z(kz)
2/* I with an error of relative order h ,
m = )
™ = g K Z(Kz) exactly,
and we may note that I zN(kz) = and I Z(*z)dz = 0. (§41.)
The same remarks as in the extensional case might be made here about the
complementary character of the types (ii), (iii) in regard to their contributions to
edge displacement or traction, when h is small.
(iv) If we follow the lines of the discussion of the extensional case, we have now
to consider the approximate allocation of a given system of edge traction among the
three types of strain.
220 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OP
The investigation is this time more complex, chiefly in consequence of the presence
of the stress nz in the permanent mode. Since, moreover, flexure is much more
important physically than extension, we shall give a fairly detailed discussion in the
next article, but in the meantime we may examine what could be done with a solution
in which, as in § 42 (iv), the principal part of the edge traction is of order zero for
each type of strain, and the parts of higher order are neglected.
Such a solution would give
N = N, +N (j \
s = s,, + s,
z = z lt j
But we see at once that we do not in this way get a perfectly general distribution of
N , S , Z , since the last equation gives / h Zdz = . A closer examination is therefore
necessary, and it will perhaps conduce to lucidity if we consider separately the three
cases of normal, tangential, and perpendicular traction.
45. Flexural forces.
(i) Normal traction.
JO I
N being of order zero, we can satisfy the conditions by taking zF, p , and g K all of this
order, but besides the terms of order zero in the stresses, it will be necessary to take
account of the terms of m which come from F and v//, albeit these are of an order one
higher. Then
o
.; =,^f + % k n(kz) . . . (i)
=** 2 F + § . (2)
Assuming these provisionally, multiply (l) by z and integrate from —h to h.
*>*&'/>* ■  ■ «
From (2), since ^ is odd in z, and ^ = for z = ± h , we get
^ = a;. 3 /r^.,F . . . (6)
In (3) the terms are of different orders ; thus, with the help of (6),
£ 3 F + J^F = . . . (7)
1 9k Z(kz) = . ... (8)
(4) and (7) define F, (6) then gives the edge value of ^ , and (5), (8) determine c/ K ,
the functions to be expanded obviously satisfying the conditions of § 40 (ii).
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 221
Further, with the values of F , ^ , g K so determined, the equations (1), (2), (3) are
all satisfied, the traction nz vanishes exactly, and the residual tractions nn ns are of
order 1.
(ii) Tangential traction.
8 being of order 0, zF and rf will again be of this order, but it will be seen that g K
is of order 1. For, making these suppositions, we have
= .^F . . . . (1)
#=^F +& .... (2)
2/x  dz
= £( Z *_7* 2 )£ 3 F + ^ + % Uk Z{kz) . . . (3)
From(l), ^F = . . . (4)
From (2), ^ f S&=  W ~ &P F + ^ • • • (5)
Differentiate this with respect to .9, and subtract from (3). Thus
" ^/_ a S ^=^ 2 ^)(^ + ^2F) + S^Z(k Z ) .. (6)
and, integrating the last from —h to h,
(4) and (7) define F, and (5) integrated from to z gives \J/. With the values so
determined, and with (6) satisfied by g K (and this is possible in virtue of (7)), the equa
tions (l), (2), (3) are all satisfied, the traction nz vanishes exactly, and the residual
tractions nn, ns are of order 1.
By combining this with the preceding case, we see that the results do in fact give a
first approximation to the solution, since the residual stresses N , S are each of an order
higher than their original given values.
The additional equation required to define g K might be found by carrying out the
process of (i) with the residual normal traction 4/xyy . The analysis would be prac
tically the same as will be given in connection with the next case,
(iii) Perpendicular traction.
Z being of order zero, we shall have to take zF and ^p of order — 1 ,' g K of order 0.
On this hypothesis, we shall write down the exact expression for nz , and the terms of
order — 1 and in 7 m and ^. We are then to have
= 2S X F
= ^ + §
(I dxl/
+ 2 ^^ + ^N(k 2 ) . (1)
2 dip
/n'\
p dn ' ^'
^ =£(*> 7, W + ^+ 2 !/k Z(kz) . . (3)
222 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF
In the first two stresses the terms of order — 1 are annulled by taking
^F = (4)
^(^JMO^F ... . (5)
Then integration of (3) from — h to h gives
^ + ^ 2 )F=4^/;> .... (6)
and from this with (3),
Another equation is required before g K is defined, but in view of the results of (i), (ii)
we can already infer that the values of F , ^ determined by (4), (6) and (5) are the
correct principal values, since the residual stresses nn , ns are of order zero. As for g K ,
we note that we cannot complete its definition by annulling the residual stress nn , for
the condition J _ h 2 = ^ zdz = is not satisfied. In order to get the remaining equation
for g K we must therefore solve by the method of (i) for the residual normal traction
4/iT t , so as to get the equation corresponding to (i) (5). The matter might be left
at this stage, but it may be interesting actually to carry out the process of balancing
the residual parts of nn and ns. By so doing we shall not only define g K , but also
obtain a second approximation to F and \^ .
We have to introduce into the solution F', >//, gj with z¥' and X of order , gj of
order 1. The equations to be satisfied are
d dh \
 2 dTJn **("> V" • • • W
_2 (fy r/y
p dn  j r ^ 2 . \ ;
=^_/^,F'+ ^ + S<7«'Z(k Z ) . . (10)
We must first find the principal value of y in terms of defined quantities.
Now from (5),
if/ = (£z 3  \hh)$.$, at the edge.
Hence, by Fourier's Theorem,
32 ,,n tV • « 1 . 3tTZ 1 . 5tt2 \
. *=  V* *■*** [ Sm 2h ~¥ sm Wi+fr sm 2h ■'•••)
Within the plate, therefore,
32/ . « 1 . 3ttz 1 . 5tt2 \
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
223
where
and y^ m has the edge value £ 2 F.
Thus
,, (2m + l)V, A
— =  — r ft I 2  sm —   — A P sin — — + . . .
dn it" \ dn 2ft 3 4 dn 2ft
the principal value of which is
Let
then
and
I67911 r?/ ■ TTZ 1 . 3tTZ , 1 . 5ttZ
— //,j,i sin —  —sin + — sin —
7T 3 2 I 2ft 3 3 2ft 5 3 2ft
ai n.\ 16/ • 7T2 1 . 3ttZ , 1 57T2
^/ft)._ sin_sin_ + _ S i 1 i
lr/3(z/h).%F
J a arc 7r" \ 3° 5 B /
if
=  1 f 75 ft^,F
7T
7." »+£+£+
Now multiply (8) by 2, and integrate from —h to A.
Hence
^F'=^y 5 ft^^F
7T S OS
and
2<7 k N(kz) = J 2Vp(z/h)  384^V f * <>,,F
(11)
(12)
(13)
(14)
(15)
This, and equation (7), define g K .
As we do not require gj, we will eliminate it from (10) at once by integrating.
Thus
'"W
Multiply (9) by 2 and integrate. Then
If z #
P i A
,^ 2 =^ F
f
and from this with the last
or, from (13),
ds\ p J n dn f 3 V 2 ds '
^F' + ^ 2 F'=^iy 6 ft/ ]
(16)
(14) and (16) give F, and ^' may be found from (9).
224 MH JOHN DOUGALL ON AN ANALYTICAL THEORY OF
If we write F for F+ F', we get from (4), (14)
ds
neglecting the term in JiF', which is of order h 2 relative to F.
This may be written
(*+3S*a*>° • • < 17 >
Similarly from (6), (16) we obtain
/„ . d a , 384 . d 1 a \— 3 /"" r/ .
We may regard (17) and (18) as the equations giving F to a second approximation.
If we combine the results of the three cases of this article, we obtain
3 <•"
3 3 F + — &,F=  JL_J f Zr7z + — f" zStfe I
'As" 4jU.A 3 ( 7 7l ffS .' 7l J
(19)
These are the equations usually referred to as KirchhofTs boundary conditions. The
extension of the more approximate conditions (17), (18) to the general case will be
given in the next article.
46. Flexural .strain wider given edge tractions. The Green's function method
for the permanent mode.
The displacement at (x f , y', z') due to tractions N , S , Z is defined in § 33, II. (i), in
terms of the work difference from the system
<£=  (3/32^) («xWx)
e = (3/32^^) (2x _ i 2 3 V 2 x _ . 2]  Wx)
From this system let the terms conveying resultant stress be removed ; the residue is
still an F strain with F = F say, and F is of order h~ s .
We have to balance F at the edge, and the edge displacements in the balanced
solution being //, q', id', the work difference required (F 1 of § 33) is
I I (Np + Sq + Zw')ds dz .
The problem is to determine p', (/, tv' as closely as is practicable.
The tractions to be balanced are
' r ' u Lwitli terms ui2 d
'Aj1,,.= (.r/r)^F .
THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
225
With tractions of these orders, of quite general form in z, the analysis of last article
would lead us to expect that in the balancing solution F would be of order — 3,
^ of order 0, and g K of order  2. But in consequence of N being simply proportional
to 2, it will be noticed that § 45 (i) (5) gives 2(/ K N(/c2) = 0, and it follows that in the
present case g K will be of order — 1 at lowest. The displacements p f , w' derived from
the strain defined by the functions g K will therefore be of order 0, and q' of order 1.
Thus we may anticipate that the first two terms of p', and the first three terms of q' and
w' will be obtained in practicable forms, i.e. independently of series associated with the
zeroes of sinh 2k.1i — 2xh.
The tractions written down above may be balanced approximately by strains F', \f/.
We require
1(*»W 8 (F + F)
+ *V'
dsdz
These are equivalent to
^(F + F') =
(^ + iU 2 )(F + F') =
(1)
which determine F', \^'.
The principal terms of the residual tractions are
j
ns _ 2 d{j/'
2fx p dit
(2)
/3(z/h)& 2 (F + ¥')
as in (iii) (12) of last article, and they may be dealt with in the manner illustrated
there.
72 / "
To balance them take strains defined by F", \f/', g K " with zF", Vy , g K " of order  1.
We must therefore have
2h^(z/h)^ 2 (F + F) =  z^F"
(2h*l P )ftz/h)» a (E + F) =  e& % If* + %
IXZ
o = hz2h*)$ 3 r+f£
2 ds dz
From these, as in last article,
384 d
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8).
+ ?</ k "N(kz)
+ 2g K "Z(«z)
ttz 1 . 3ttZ
(3)
(4)
34
226
MR JOHN DOUXIALL ON AN ANALYTICAL THEORY OF
whence
'!£= Vf3(z/h)W" +  ( I z 8 * \hh ] .■'.( I
(5)
We are now left with tractions «b , ns of order zero. In the solution balancing these,
¥'" will be of order — 1 , >//" of order 2 , g"' of order . We do not think it worth
while to write down the equations defining F w , but it should be noticed that they can
be found explicitly. In fact, although the residual tractions nn , ns with which we are
now dealing are partly defined by k series, the integrals I znndz and / Znsdz will be
found to vanish to the order concerned so far as they come from these series, in virtue
of the relation ^gJ"^{ K z) = , which follows from (iii) (7) of last article.
The functions g K " give terms of order zero in p', w' '.
p' terms of orders —2,1
q' 2,1,0
Hence, including in
w
we have
p'=4(l^(F + F + F")
q =  4(1  cr)a.(F + F' + F" + F") +
3,2,1
42<r
r ( 6 )
w = 4(1  <r)(F + F' + F" + F") + 2(crr  /r)y 2 (F + F') .
The value of J to a second approximation is
gl =  h^(z/h)S 2 (F + F)  1(1 Z 3 _ 1/^^ 2 (F + F)
(7)
and
4n is given by (5).
The function Y x which (§33) defines the permanent solution is
//
(Np' + Sq + Zw')ds dz
(8)
For the case of a solid or hollow circular plate all the quantities in the righthand
members of (6) can actually be calculated, and we thus obtain the solution for normal
traction to a second approximation, and for tangential or perpendicular traction to a
third approximation, in a form, moreover, applicable without modification or addition
even when the given edge stresses are discontinuous.
We conclude by deducing the equations corresponding to Kirchhoff's boundary
conditions to a second approximation. (They might be found to one order higher in
the case of vanishing normal traction.)
We suppose that the given tractions N , S are of the same order in h, and that Z is
of an order one higher. Any case may be reduced to combinations of cases satisfying
this condition.
THE EQTJILIBKIUM OF AN ISOTROPIC ELASTIC PTATE.
227
4(lo)
ds
0)
If we write F for Fq + F' + F", then it follows from (6), (7), (8) that the terms of
two lowest orders in Fj. are given by
dnj h ds) h J h
+ (d**_Lf\r WfK ,/ h)6d .
\ds an p as I J h
As in the extensional case, § 43, the integral with respect to s may be modified by
cZF
means of integration by parts so that only F and ^ appear under the integral sign.
Write
/* zNdz=N 1} j h zSdz=B 1 , f Zdz = Z
J h J h J h
f l 2h 2 (3(z/h)8dz=S 2 .
J h
Then if N t , S x , Z , S 2 are continuous functions of s ,
Hence any two systems of traction will give the same permanent mode to .a second
approximation, provided the values of ^i + jj and ~^i +Zu+ ^ JL
are the same for the two systems.
Now for the system F x , § 44 (i),
(10)
ds
ds
ds
Nj =  i M 3 ^iFi , Si =  I/AV.F, , Z =  £ M^i ,
S, =  I_^ 3 ^ 75 M 2 F, , § 45 , (iii) (13).
3 7T
(12)
Thus, with #! , $., , $, defined as in § 44 (i), the boundary conditions are
**(w + £»»£v,)*.+t
r/.s
ds p
ds
ds p j
(13)
When only the principal terms are retained, these reduce to Kirchhoff's conditions.
If S x or S 2 is discontinuous at any point of the edge, integrated terms will appear
in equation (11), as in the extensional case, § 43. Thus, if the normal couple S x be
discontinuous at a point P (s = s'), there will appear on the right of (11) a term
F(s')  S :
s = s +
The method of dealing with such a discontinuity in any actual problem is obvious, for
by (9) its effect is the same as that of an element  S x [ of shearing traction applied at
P, a result which on the ' elastic equivalence ' theory may easily be obtained by a
trifling modification of the process by which Thomson and Tait reconciled the con
ditions of Kirchhoff and Poisson.
228 THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE.
Addition to Paper by J. Dougall on —
" An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate."
(Note added May 20, 1904.)
The kindness of one of the referees enables me to supply the following references
to recent work bearing on the subject of the paper.
(a) J. H. Michell, in a paper " On the direct determination of Stress in an Elastic
Solid, with application to the Theory of Plates," Proc. Lond. Math. Soc, vol. 31^
1899, shows how the stresses might be found without previous determination of the
displacements. In the case of the stress zz or R, he finds that y 4 R is a given function
in the body of the plate, while R and d^Kjdz are given on the faces. If we neglect the
conditions at the edge, which have practically no influence on the result, a value of R
satisfying these conditions can be found, in terms of Fourier integrals for instance. Mr
Michell does not determine R — this has been done in the present paper — but proceeds
to deduce the forms of the remaining stresses, and the differential equation for the
normal displacement of a point on the mid plane. One special case of normal force is
worked out to a first approximation, and Lagrange's equation for this case deduced.
For the conditions at the edge, reference is made to the ordinary Thomson
Boussinesq theory, which uses the principle of equipollent loads.
(b) L. N. G. Filon, " On an approximate solution for the bending of a Beam of
rectangular cross section under any system of Load, with special reference to points of con
centrated or discontinuous Loading," Phil. Trans. R. Soc. Lond. (Sec. A), vol. 201 (1903).
Dr Filon's solution applies to a beam in which the ratios of breadth to depth, and
of depth to length, are both small. The axis of z being taken in the direction of the
breadth, the stress zz is taken as negligible, and equations are deduced for the mean
values, across the breadth, of the displacements u, v. These equations are the same as
equations (90), page 182 of the present paper, with the body force null. In order to see
the reason of this from our standpoint, we may notice that the assumption that 22
vanishes eliminates all the solutions of what we have called the dilatational transitory
type, and that taking the mean of the displacements eliminates all the flexural solutions,
as well as the rotational transitory solutions.
As regards the conditions at the ends, the beam is treated as a long rod.
It may be of interest to remark that the results of § 43 above furnish the data for
a more approximate treatment of the problem on the lines followed by Dr Filon.
(c) A note appended to a paper by Professor Lamb in Proc. Lond. Math. Soc,
vol. 34 (1902), pp. 283, 284, contains a solution of a special case of the problem of
face traction.
(d) In connection with existence theorems relating to the elastic equations, reference
should be made to the work of Italian elasticians, as Somigliana, Lauricella, and
Tedone  PRESENTED
9 FZB. jyui.
The Transactions of the Royal (Society of Edinburgh will in future be Sold
at the following reduced Prices : —
Vol.
VI.
l'riee to the
Price to
i
Vol.
Price to the
Price to
Public.
Fellows.
£0 9 6
Public.
Fellows.
£0 11 6
XXXIX. Part 1.
£1 10
£1 3 '<
VII.
18
15
Part 2.
19
14 6
VIII.
17
14
Part 3.
2 3
1 11
IX.
1
17
Part 4.
9
7
X.
19
16
XL. Part 1.
1 5
19 
XI.
14 6
12
„ Part 2.
1 12 6
1 5 6
XII.
14 6
12 i
„ Part 3.
1 6
19 6
XIII.
18
15
XIV.
1 5
110 1
XV.
1 11
1 6
XX. Part 1.
18
14
XXII. Part 2.
10
7 6 j
Part 3.
1 5
1 1
XXVII. Part 1.
16
12
Part 2.
6
4 6
Part 4.
1
16
XXVIII. Part 1.
1 5
1 1
Part 2.
1 5
110;
Part 3.
18
13 6
XXIX. Part 1.
1 12
1 6
Part 2.
16
12
XXX. Part 1.
1 12
1 6
Part 2.
16
12
Part 3.
5
4
Part 4.
7 6
5 8
XXXI.
4 4
3 3
XXXII. Part 1.
1
16
Part 2.
18
13 6
Part 3.
2 10
1 17 6
Part 4.
5
4
XXXIII. Part 1.
1 1
16
Part 2.
2 2
1 11
Part 3.
12
9 6
XXXIV.
2 2
1 11
XXXV.*Part 1.
2 2
1 11
Part 2.
1 11
1 3 6
Part 3.
2 2
1 11
Part 4.
1 1
16
XXXVI. Part 1.
1 1
16
Part 2.
i it; 6
1 7 6
Part 3.
1
16
XXXVII. Parti.
1 14 6
1 5 6
Part 2.
1 i
16
Part 3.
16
12
Part 4.
7 6
5 8
XXXVIII. Parti.
2
1 10
Part 2.
1 5
19
Part 3.
1 10
1 3
Part 4.
7 6
5 8
* Vol. XXXV., and those which follow, may be had in Numbers, eacb Number containing
a complete Paper.
December 1902. — Volumes or parts of volumes not mentioned in the above list are not for the present on sale to the
public. Fellows or others who may specially desire to obtain them must apply direct to the Society. As the Society
reprints from time to time parts of its publications which have become scarce, the absolute correctness of this list
cannot be guaranteed beyond this date.
TRANSACTIONS
OF THE
ROYAL SOCIETY OF EDINBURGH.
VOLUME XLI. PART II.— FOR THE SESSION 19045.
CONTENTS.
Page
IX. On the Measurement of Stress by Thermal Methods, with an Account of some Experiments
on the Influence of Stress on the Thermal Expansion of Metals. By E. G. Cokbr, M.A.
(Cantab.), D.Sc. (Edin.), F.E.S.E. ; Assistant Professor of Civil Engineering, M'Gill
University, Montreal. (With Two Plates), . . . . . .229
(Issued separately 2nd September 1904.)
X. On the Spectrum of Nova Persei and the Structure of its Bands, as photographed at Glasgoiv.
By L. Becker, Ph.D., Professor of Astronomy in the University of Glasgow. (With
Three Plates), .......... 251
(Issued separately 94ii September 1904.)
XI. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part I. Structure of the
Resting and Dividing Corpuscles. By Thomas H. Bryce, M.A., M.D. (With Five
Plates), .......... 291
(Issued separately 19th November 1904.)
XII. The Action of Chloroform upon the Heart and Arteries. By E. A. Schafer, F.R.S., and
H. J. Scharlieb, M.D., C.M.G., . . . . . . .311
(Issued separately lith December 1904.)
XIII. Continuants resolvable into Linear Factors. By Thomas Muir, LL.D., . . . 343
(Issued separately 13th January 1905.)
XIV. The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D. Falconer, M.A.,
B.Sc. (With a Map), ......... 359
(Issued separately 9th June 1905.)
XV. On a New Family and Twelve New Species of Rotifera of the Order Bdelloida, collected
by the Lake Survey. By James Murray. (With Seven Plates), . . . 367
(Issued separately 3rd March 1905.)
XVI. The Eliminant of a Set of General Ternary Quadrics. — (Part III.) By Thomas Muir, LL.D., 387
(Issued separately 15th April 1905.)
XVII. Theorems relating to a Generalization of Bessel's Function. By the Rev. F. H. Jackson, R.N., 399
(Issued separately 18th April 1905.)
XVIII. On Pennella bal&uopterae : a Crustacean, parasitic on a Firmer Whale, Balajnoptera
musculus. By Sir William Turner, K.C.B., D.C.L., F.E.S. (With Four Plates), . 409
(Issued separately 2Qlh May 1905.)
XIX. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part II. Htematogenesis.
By Thomas H. Bryce, M.A., M.D. (With Four Plates), . . . .435
(Issued separately 6th May 1905.)
XX. Supplement to the Lower Devonian Fishes of Gemunden. By R. H. Traquair, M.D.,
LL.D., F.R.S. (With Three Plates), .... . 469
(Issued separately 13th May 1905.)
EDINBURGH:
PUBLISHED BY ROBERT GRANT & SOK 107 PRINCES STREET,
AND WILLIAMS & NORGATE, 14 HENRIETTA STREET, CO VENT GARDEN, LONDON.
MDCCCCV.
Price Twentynine Shillings and Sixpence.
( 229 )
IX. — On the Measurement of Stress by Thermal Methods, with an Account of some
Experiments on the Influence of Stress on the Thermal Expansion of Metals.
By E. Gh Ooker, M.A. (Cantab.), D.Sc. (Edin.), F.RS.E. ; Assistant Professor of
Civil Engineering, M'Gill University, Montreal. (With Two Plates.)
(MS. received April 11, 1904. Read June 6, 1904. Issued separately September 2, 1904.)
CONTENTS.
1. Introduction
2. The Thermal Effect of Tension and Compres
sion Stress ......
3. The Thermal Expansion of Brass and Steel
under Tension Stress ....
4. The Behaviour of Iron and Steel under Tensile
Stress .......
5. The Relation of Stress to Strain and Thermal
Change in Tension Members
PAGE
229
231
232
238
241
PAGE
6. The Relation of Stress to Strain and Thermal
Change in Short Compression Members . 243
7. The Variation of Compression Stress in a Long
Compression Member ..... 247
8. The Variation of Stress in the Cross Section of
a Beam 248
9. Conclusion ....... 250
1. Introduction.
In the determination of the effects of stress upon different materials, the investigator
has several methods of attack open to him, each of which has its own particular
advantages. In the great majority of cases the material under investigation obeys
the generalised Hooke's law, and the effects of a stress are therefore most easily followed
and measured 'by observations of the strains produced. The strains being usually
exceedingly minute, it is necessary to magnify them sufficiently to allow of accurate
measurement. To this end many instruments have been devised for measuring the
Strains obtained by the action of different stresses, and in fact the great majority of
our experimental knowledge has been obtained in this way. The application of polarised
light to the determination of stress was first suggested by Brewster,* and he applied it
to many problems, particularly the determination of the neutral axis of a glass beam.
Neumann,! with a full knowledge of the work of Brewster, developed a theory of the
analysis of strain by polarised light, and Maxwell J also independently developed a
theory. A third method, which has assumed great prominence in recent years, is the
microscopic examination of metals under stress, as developed by Ewing, and Kosenhain,
and others.
The present paper is mainly concerned with the measurement of stress by the
temperature changes produced, a subject to which attention was first drawn by Weber,§
who found that when a wire was stretched suddenly a thermal effect was produced,
* Trans. R.S.E., vol. iii.
t Abhandluugen der k. Akademie der Wissenschaften zu Berlin, 1841.
t "On the Equilibrium of Elastic Solids," Trans. R.S.E., 1853.
 "On the Crystalline Structure of Metals," Trans. R.S., 1900.
§ Poggendorffs Annalen, Bd. xx., 1830.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 9). 35
230 PROFESSOR E. G. COKER ON
which he attributed to an internal cause. He deduced a theoretical formula in which
the change of temperature t was shown to bear a linear relation to the stress p and to
the coefficient of expansion Jc, which may be expressed in the form t = c k p, where c is
a constant. He proved from his experiments that a wire when stretched within its
elastic limit is lowered in temperature, and when compressed has its temperature raised.
Duhamel * investigated the modifications which became necessary in Poisson's elasticity
equations, when allowance is made for change of temperature. The subject was placed
on a sound basis by Lord Kelvin, t who deduced from the laws of thermodynamics the
general equations of thermoelasticity. He showed that the thermal effect H produced
by stresses p xy can be expressed in the form
3^~dt xv
where t is the temperature, J is Joule's equivalent, and e xy is the strain corresponding
to the stress p xy .
The general conclusions deduced were that " cold is produced whenever a solid is
strained by opposing, and heat when it is strained by yielding to any elastic force of its
own, the strength of which would diminish if the temperature were raised, but that, on
the contrary, heat is produced when a solid is strained against, and cold when it is
strained by yielding to, any elastic force of its own, the strength of which would increase
if the temperature were raised."
These conclusions were experimentally verified by Joule, J who showed that the
thermal changes produced by stretching and compressing metals, timber, etc., and by
the deflection of helical springs, was proportional to the stress applied, and obeyed the
Thomson law. The change of temperature was measured by thermoelectric couples
composed of iron and copper wires, either pressed against the specimen or inserted in
holes drilled into them ; a galvanometer was placed in circuit with the thermoj unction
to indicate the change of temperature ; to calibrate the galvanometer the test specimen
was plunged into water, at a known temperature, to within a short distance of the
junction.
Edlund § applied the methods of Joule to the determination of the effects of stress
on wires, and instead of an ironcopper junction he used crystals of bismuth and
antimony cut to a cylindrical form, and the cut ends were pressed against the wire, so
that no variation in the thermoelectric power was possible, as might be the case if the
natural cleavage plans were used. His results amply verify Joule's earlier work, and
he, moreover, obtained an approximately correct value of the mechanical equivalent of
* " Memoire sur le calcul des actions moleculaires developpees par les changements de temperature dans le corps
solides," — Memoires . . . par divers savans, vol. v., 1838.
t " On the Dynamical Theory of Heat," Trans. B.S.E., 1851.
X "On some ThermoDynamical Properties of Solids," Trans. R.S., 1853.
§ "Untersuchung uber die bei voluniveranderung fester korper entstehenden warmephanomene, sowie derein
verhaltniss zu dabei geleisteten mechanischen arbeit," Pogg. AnnaL, vol. cxiv., 1861, and "Quantitative bestimmung
der bei voluniveranderung der metalle enstehenden warmephanomene und die mechanischen warmeaequivalents,
unabhiingig von der inneren arbeit des metalls," Pogg. Annal., vol. cxxvi., 1865.
THE MEASUREMENT OF STRESS BY THERMAL METHODS. 231
heat by a direct application of the Thomson formula. In a recent valuable paper by
Turner * the methods of Joule and Edlund are substantially followed, and a detailed
account is given of experiments on tension stress for metal bars of a size needing a
modern testing machine for the stressing agent.
2. The Thermal Effect of Tension and Compression Stress.
In the present paper the main object of inquiry is tension and compression, and
for these stresses it is easy to deduce a simple form of the relation between the change
of temperature and the stress from the equations of thermodynamics.
If I be the length of unit mass of a rod subjected to a compression stress of
intensity p, which shortens the bar by an amount dl, and E be the initial intrinsic
energy of the bar and dH. the amount of heat developed by the compression,
we have
dE + pdl = Jdh
'{(D,**(fH
= J<p f dp + t p dt \ . . . . . . (1)
where Pt is written for the more cumbersome symbol (%) •
Now the alteration of length is a function of the pressure and temperature, and
hence we have
dl= dp + dt.
op at
Therefore
dE = (jp t  P ^) dp + (jt,  v gj) dt
a perfect differential, whence
d7\ J P<PcTj) = dp{ Jt »Pdt
giving
j(dt z _dp i \J_l
\dp dt/ dt
Equation (l) can be written in the form
t t p+ t
a perfect differential also, from which we obtain
_ pt ^ /dtp _ dp,\ = 1 dl
t ' \dp dt) Jdt
Now if a body be compressed adiabatically
t p dt +p t dp =
dt p t t dl 1
or — = £; = .
dp t v J dt t p
* «
ThermoElectric Measurement of Stress," Trans. Amer. Soc. C.E., 1902.
232 PROFESSOR E. G. COKER ON
If a be the coefficient of expansion of the bar for unit increase of temperature, we can
write this in the form
dt_tal
dp~Jt p '
and for small changes of pressure and temperature we can write the equation in
the form
F tal
where the sign of At depends upon the signs of a and Ap, since all the other quantities
are essentially positive.
For metals « is in general positive, and hence a compression stress will raise the
temperature, while a tensional stress will lower it. Strictly speaking, the equation only
holds for infinitesimal changes of p and t, and it is therefore essential to show what
limitations, if any, are to be imposed in its application to bodies under great ranges
of stress. The effect of a varying load upon the specific heat of a body has not been
determined, so far as 1 am aware, but it is unlikely to differ by an appreciable amount
from the specific heat at atmospheric pressure. The coefficient of expansion when the
stress is varied may change to a small extent, and experiments were made by Joule *
to examine the effect of stress upon the expansion of various timbers, and he found
an increase of expansibility with tension. As far as I am aware, there are no
experiments showing what effect stress has upon the thermal expansion of metals ;
and as the coefficient of expansion enters into the fundamental equation, a special investi
gation was made of the thermal expansion of brass and steel under different tension loads.
3. The Thermal Expansion of Brass and Steel under Tension Stress.
The general arrangement of apparatus adopted is shown in fig. 1, in which A is
the standard of a small singlelever testing machine provided with a weighbeam B
and shackles C D for securing the testpiece E in position. The loading of the
speoimen is effected by suspending dead weights F from the end of the beam, and the
maximum load which could be safely applied was 175 pounds. The ratio of the arms
of the lever was 20 to 1, and hence the maximum stress obtainable was 3500 pounds.
In order to carry the experiments past the elastic limit, it was necessary to have
a specimen of very small section ; and on account of the difficulty of maintaining a
solid specimen at a uniform temperature, and at the same time observing the change
of length, the specimens were chosen of seamless drawn tube, very uniform in diameter,
thereby permitting the outside being turned in a lathe to a manageable section. The
ends of the tube were soldered into thick ferrules G, having side tubes H for the
insertion of thermometers, and inlet and outlet tubes I F were provided, connecting
to a pipe system J, in which water could be circulated at any desired temperature.
The circulation was effected by a small centrifugal pump K driven by an eleetric
* " On some ThermoDynamic Properties of Solids," Phil. Trans., 1859.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
233
motor L, and provided with an extension barrel M, heated by a gas flame N. The
rotation of the pump caused water to be drawn in at the eye of the pump from the
pipe V, and to be discharged through the perforated partition into the encircling
heater M. From thence it flowed through the rising pipe J and the specimen, as
indicated by arrows. A very vigorous circulation was maintained, and the pipes were
thickly lagged, so that there was practically no difference of temperature between the
end points of the tube under measurement. The readings of the thermometers were
found to be practically coincident at all temperatures, and therefore in the tabulated
results one set of temperature readings is omitted. The alteration in the length of
M^^
the tube with increase of temperature was determined by aid of a Ewing extensometer
of the original pattern. The instrument (fig. 2) consists of a pair of clips A B, secured
to the tube by set screws C ; the upper clip carries a frame D, provided with a
calibrating screw P and a reading microscope F focussed on the edge of a thick wire
W carried by the lower clip. Any alteration in the length of the tube causes a
movement of the wire relative to the microscope, and by aid of a glass scale inside
the eye piece the alteration in length can be determined. The dimensions of the
instrument were such that a movement of '00002 inch corresponded to one unit
of the scale, and the micrometer screw enabled a calibration of the scale to be effected
while the instrument was attached to the specimen — a great advantage with an
extensometer. The construction of the apparatus permitted a thick layer of lagging K
to be applied. It was proposed at first to surround the specimen by a waterjacket,
fed from the circulating system, to ensure the temperature of the tube being absolutely
uniform throughout, but this was not done, as the lagging was found to be very
efficient, and the extra complication did not seem warranted, particularly as only
234
PROFESSOR E. G. COKER ON
comparative readings were required. The thermometer bulbs I were inserted in the
main tubes, and secured by flexible rubber joints J, # as indicated.
In carrying out a test the centrifugal pump was first set in operation, and when
the readings of the extensometer became steady, the gas flame was applied to the
extension barrel of the pump, and the readings were taken, at intervals, of the
temperature and the extension, until a temperature of about 180° Fahr. was reached
A new experiment was then begun with a different loading. The testing machine used
was not sufficiently powerful to overstrain the tubes used, as these latter were
exceptionally hard, owing to their manner of production ; and for the experiments
with a permanent overstrain the tubes were taken out and stressed in a Kiehle
testing machine of 60,000 pounds capacity.
The results of the tests on the brass tube are shown in Table I., from which it will
* " On a flexible Joint for securing Thermometer and like Stems and Tubes," E. G. Coker, Physical Review, 1903.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
Table I.
Brass specimen. — Internal diameter = '378 inches.
External diameter = "525 ,,
Area =0  1042 square inches.
235
Stress in lbs.
per sq. inch
)
960
9,597
19,194
28,791
33,590
Load, lbs. .
!
100
1,000
2
000
3,000
3,500 \
Yginch
overstrain
•^inch
overstrain
^inch
overstrain
Temp. Fahr.
Reads, a
Reads, a
Reads
. A
Reads, a
Reads, a
Reads
A
Reads
. A
Reads, a
70
20
23
22
21
20
22
22
22
75
20
23
22
21
20
22
22
22
21
21
21
21
21
21
21
24
80
41
44
43
42
41
45
43
46
23
 22
21
24
21
22
23
21
85
64
66
64
66
62
67
66
67
23
21
22
19
20
19
21
25
90
87
87
86
85
82
86
87
92
22
21
20
21
20
23
22
21
95
109
108
106
106
102
109
109
113
22
21
21
21
19
21
22
21
100
131
129
127
127
121
130
131
134
22
21
23
21
19
21
22
22
105
153
150
150
148
140
151
153
156
 22
21
22
 22
20
21
22
23
110
175
171
172
170
160
172
175
179
21
21
21
23
20
21
21
24
115
196
192
193
193
180
193
196
203
23
21
21
21
21
22
23
21
120
219
213
214
214
201
215
219
224
22
21
21
23
20
21
21
24
125
241
234
235
237
221
236
240
248
22
21
21
24
21
22
21
23
130
263
255
256
261
242
258
261
271
22
21
21
24
22
22
20
25
135
285
276
277
285
264
280
281
296
22
21
21
21
23
22
22
24
HO
307
297
298
306
287
302
303
320
21
21
22
18
24
 22
22
23
145
328
318
320
324
311
324
325
343
22
21
22
22
25
21
26
26
150
350
339
342
346
336
345
351
369
22
21
 22
22
23
22
20
22
155
372
360
364
368
359
367
371
391
22
21
21
23
23
24
24
26
160
394
381
385
391
382
391
395
417
22
22
22
24
23
23
24
26
165
416
403
407
415
405
414
419
443
22
23
21
20
24
25
22
28
170
438
426
428
435
429
439
441
471
21
24
25
24
23
25;
23
24
175
459
450
453
459
452
464
464
495
21
24
21
23
25
24
25
180
480
479
474
482
477
480
489
—
236
PROFESSOR E. G. COKER ON
be seen that the extension for all the different loads below the yield point of the
material are practically constant. This is shown graphically by fig. 3, in which the
readings are plotted as ordinates, with the temperatures as abscissae. The curves for
stresses below the yield point are very nearly straight lines, with the exception of
No. IV., when a load of 3000 pounds was applied, corresponding to a stress of 28,790
pounds per square inch. We may neglect the small deviation there shown, since at an
increased stress of 33,570 pounds per square inch it disappears, and we may assume
that the expansion is practically linear. The mean value of the coefficient of expansion
for these five different experiments corresponds to a linear expansion of '00001953 for
1° centigrade, the maximum deviation therefrom being slightly less than 1 per cent.
For the overstrained tube the coefficient of expansion was greater, the values
obtained being as follows :
Table II.
Total Permanent Extension.
Inches.
Coefficients of Expansion per
1° Centigrade.
i
16"
1
1
¥
•00001963
•00002004
•00002121
A similar experiment upon a steel tube having very thin walls was made, and the
results are given in Table III.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
237
Table III.
Steel specimen. — Internal diameter =0 "406 inches.
External diameter =0 '482 inches.
Area =0*067 square inches.
Stress in lbs.
per sq. inch
I 1,493
14,926
29,852
35,824
37,157
Load, lbs.
I 100
1,000
2,000 J
2,400
■Jin. overstrain
2,500
£in. overstrain
Temp. Fahr.
Beads
A
Reads. a
Reads. a
Reads. a
Reads. a
70
12
13
11
12
12
75
12
13
11
12
12
14
13
12
12
12
80
26
26
23
24
24
1
12
12
13
11
10
85
38
38
36
35
34
12
12
13
13
12
90
50
50
49
48
46
12
13
12
13
11
95
62
63
61
61
57
13
12
11
12
12
100
75
75
72
73
69
11
11
12
12
13
105
86
86
84
85
82
13
12
13
12
11
110
99
98
97
97
93
14
13
13
12
13
115
113
111
110
109
106
13
11
12
13
13
120
126
122
122
122
119
12
12
13
11
13
125
138
134
135
133
132
IS
12
13
12
11
130
151
146
148
145
143
13
11
12
13
12
135
164
157
160
158
155
12
13
12
14
16
140
176
170
172
172
171
13
12
13
11
11
145
189
182
185
183
182
13
13
14
13
12
150
202
195
199
196
194
14
11
 12
12
14
155
216
206
211
208
208
12
12
13
12
13
160
228
218
224
220
221
13
12
14
15
13
165
241
230
238
235
234
13
13
12
15
13
170
254
243
250
250
247
16
13
15
14
175
259
263
13
265
15
260
180

—
276
280
—
TRANS. RO
Y. SOC.
EDIN., i
IOL. XLI. PART
IT. (NO. 9).
36
238 PROFESSOR E. G. COKER ON
This differed somewhat from the brass tube previously experimented upon, in showing
a practically uniform coefficient of expansion under every load below and above the yield
point (fig. 4). The mean value of the coefficient Avas found to be '00001121, the maximum
deviation being for the two lowest loads, and amounting to nearly 2*4 per cent., while
for the remaining loads the deviation was less than 1 per cent. The general accuracy
of the results was checked by comparison with the known values of the coefficient of
expansion at atmospheric pressure, and the agreement is sufficiently close to make it
clear that the observations were accurately taken, having regard to the fact that the
seamless tubes experimented upon had been subjected to exceptional treatment in the
process of manufacture, and were probably in a very different physical condition from
solid bars of the same material rolled or cast in the ordinary way.
We may conclude from these experiments that there is practically no difference ' in
the linear expansion of brass and steel within the range of stress up to the yield point
of the material, and that for brass there is probably an increase in the coefficient beyond
the yield point, but that there is no increase for steel. In this part of the work I was
greatly assisted by Mr Charles M'Kergow, Demonstrator in the Civil Engineering
Department, M'Gill University, who kindly undertook the major part of the work of
observation, and who also rendered me very able assistance in the experimental work
detailed below.
4. The Behaviour of Iron and Steel under Tensile Stress.
In applying the thermal method of measuring stress, the most convenient arrange
ment of apparatus consists of a thermal junction or pile of the necessary delicacy con
nected to a galvanometer giving a sufficiently wide range of readings for the small
difference of temperature produced. In nearly all testing laboratories the presence of
iron in large quantities makes it necessary to choose a galvanometer which is not
influenced by the proximity of iron, and this was especially necessary in the present
case, since in the M'Gill University testing laboratory the main testing machines have,
in the course of time, become magnetised, owing to the subsidiary mechanism being
operated by electric motors. These difficulties are easily overcome by the use of a
D'Arsonval galvanometer, the field of which is very uniform, even in the neighbourhood
of large masses of feebly magnetised iron. The galvanometer coil was specially wound
for me by Dr Tory, and was of approximately the same resistance as the thermopile
used in the majority of the experiments, so that the arrangement was as sensitive as
possible. To avoid shortcircuiting of the pile when in contact with the metal specimen
under test, it is convenient to insulate it therefrom, and I have found a thin sheet of
paper, as suggested and used by Joule, the most convenient. The connections of the
galvanometer to the thermopile were made by soldered joints, which were afterwards
wrapped in paper to insulate them from one another, and then tied together and lagged
with cotton wool.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
239
In making observations with a galvanometer provided with a moving coil of con
siderable weight, such as that of the D'Arsonval type, the indications may not be a
faithful record unless certain precautions are observed, for the galvanometer does not
take up its position of equilibrium at once, and therefore any error due to this lag will
make a considerable difference in the results. It was found experimentally that the
time rate of loading could be so determined by trial that the reading of the galvanometer
was very approximately a maximum for the stress at any given instant, and by making
special experiments for each bar, the rate at which the loading ought to be applied
could be easily determined. As an example of the accuracy with which the loading
could be applied to keep step with the galvanometer, reference may be made to the
results obtained from a steel bar of rectangular section, having a breadth of 0'86 inches
and a thickness of 0*315 inches, which was loaded at a uniform rate until a maximum
of 4000 pounds was reached, corresponding to a stress of 14,760 pounds per square
inch.
The readings obtained were as follows : —
Table IV.
Load, pounds.
Time in seconds.
Observed Reading.
Corrected Reading.
2000
25
40
423
3000
45
70
771
4000
60
98
1112
4000
90
90
1082
4000
110
80
998
4000
135
70
913
4000
165
60
823
4000
200
50
725
4000
260
40
634
4000
340
30
560
4000
410
25
481
Notes. — Scale distance, 83 inches: 1 division =  5 inches on scale.
Resistance of thermopile and leads = 6  07 ohms.
Resistance of moving coil of galvanometer = 5*2 ohms.
Temperature 68° Fahr.
From which it will be seen that as soon as the loading reached a maximum, the readings
also attained a maximum, and then began to decrease. The ascending portion of the
240 PROFESSOR E. G. COKER ON
curve showing the relation of the thermal change to the stress is, however, influenced
by losses due to conduction and radiation, and hence it is necessary to correct for these
in order to obtain a correct relation. If it be allowed that the lag of the galvanometer
is not a factor in this correction, the required result may be obtained as follows. Let
be the diminution of temperature per second due to the application of a stress
increasing uniformly with the time, and let 6 be the actual difference of temperature
at any time t from the commencement of the application of stress, then 0<0 t, since
there is a loss due to conduction and radiation, depending on the difference of
temperature between the specimen and surrounding bodies. The loss due to this
cause can be very approximately determined by observation of the subsequent readings
when the application of stress has ceased, and it was found in all cases that the loss
was very accurately proportional to the first power of the difference of temperature.
In an interval of time dt, therefore, the diminution of temperature for a tension specimen
under uniformly applied stress will be 6 dt — k0'dt, where k is a constant to be
determined. The actual decrease of temperature in the time dt is rrdt.
Hence
an integrating factor of which is obviously e kt .
Hence
S'O^oJs'dt + c
k
or = o /fc + ce*'.
To determine the constant c we have the condition that is zero at the commencement
of the application of the load ; hence c = — 6 /k, and we have
or O = &0/(1 €*')
Now the denominator can be expanded provided the value of the variable t is such
that the expression in the bracket remains convergent, and it is evident if kt/2<l i.e.
t<C 2 /k this condition will be satisfied. Hence we obtain as a sufficiently near
approximation
and since & is a very small quantity, this reduces to
Now t is the actual decrement of temperature D t due to the stress up to the time t,
THE MEASUREMENT OF STRESS BY THERMAL METHODS. 241
and is the observed value D A . Hence the observations for the ascending part of the
curve must be corrected by the formula
D,= D a (l+).
The value of Jc is determined in each case by the second part of the curve ; and in the
example shown and in all others described in this paper, it is of the exponential type.
In the present example the value of the deflection D at any time after the loading
ceased was found to be
 0045*
D = 98e
where t is the time in seconds from the cessation of the load. The curve showing the
readings corrected for the radiation loss during the loading is shown dotted in fig. 5.
There is also a correction for the change in resistance of the galvanometer coil and
leads, owing to the change in the temperature of the room. The testing laboratory
was very favourably situated in this respect, as its temperature rarely varied more
than two or three degrees, and hence this correction was unnecessary. A further
correction might be made since the current strength i in the thermopile circuit, and
therefore the deflection of the galvanometer, =£—5— and where 4> is the angle turned
° COS <p ' °
through by the moving coil, but in all cases the deflection was so small in comparison
with the distance of the scale from the moving coil and mirror attached thereto that
the correction was negligible.
5. The Relation of Stress to Strain and Thermal Change in
Tension Members.
The variation of strain with regard to tension stress follows a linear law very
approximately over a certain range in the case of most metals, and in the case of iron
and steel this linear relation holds for a considerable part of the whole range of stress
up to rupture. This is easily shown by delicate extensometers, such as those devised
by Unwin, Ewing, Martens and others. It becomes of importance to determine what
is the relation of the thermal change to stress and to strain. The only previous
experiments of which I am aware are those of Turner,* who has experimented upon
the relation of thermal change to stress ; and from the known properties of iron and
steel as regards strain, he has deduced from his results that " the thermal limit of
proportionality is lower than what is considered the true primitive elastic limit of the
metal." He suggests that there exists from the thermal point of view a welldefined
range of almost perfect elasticity, beyond which " there is a considerable, in fact nearly
equal, range of imperfect. elasticity, before reaching the limits of apparent elasticity of
shape." This is a matter of considerable importance in regard to the question of
repeated stress, since if this is so, it may have an important bearing on the results of
Wohler and others.
* hoc. cit.> ante.
242
PROFESSOR E. G. COKER ON
In order to test the truth of this, several experiments were made on bars in tension,
using a thermopile for measuring the cooling effect caused by the stress applied, and
an extensometer to determine the strain. This latter was of the usual Unwin pattern,
except that the metal distancepieces were replaced by mahogany rods, previously soaked
in paraffin wax,' and suitably capped. This precaution renders negligible any error due
to any slight changes of temperature, as the coefficient of expansion of mahogany is
extremely small. The steel bar quoted above, for which the correction factor for
radiation and conduction had already been determined, may be quoted as an illustration.
The load was applied as uniformly as possible in a Buckton testing machine at a uniform
rate of 4000 pounds per minute, and the galvanometer and extensometer readings were
taken at each interval of 1000 pounds. The following readings were obtained : —
Table V.
Extensometer
Observed
Corrected
Galvanometer
Time, seconds.
Load, pounds.
Reading.
Galvanometer
Reading.
Reading.
7„=0045.
16
1,000
25
259
30
2,000
200
105
50
534
44
3,000
305
105
80
879
60
4,000
410
105
105
1192
75
5,000
520
105
133
1554
91
6,000
625
100
163
1960
7,000
725
100
183
120
8,000
825
102
205
2604
9,000
927
103
228
150
10,000
1,030
100
250
3344
11,000
1,130
120
268
180
12,000
1,250
287
4032
195
13,000
300
4316
14,000 1
Went off scale at
Galvanoniet
er reading
13,600 lbs.
went off
scale
Note. — Distance of scale from mirror of galvanometer = 6' 11 \".
The galvanometer used in all the experiments mentioned in this paper was provided
with a coil of resistance 5'2 ohms, and having about 300 turns; the thermopile was
THE MEASUREMENT OF STRESS BY THERMAL METHODS. . 243
approximately square in section, with 31 couples, and had a resistance of 5*55 ohms.
The short connecting wires or leads had a resistance of 0"52 ohms. In a few experi
ments, which are specially noted, a linear pile was used of 10 couples and of 0'18 ohms
resistance, and also long connecting wires or leads of 1*81 ohms resistance were used in
some cases. A plot of these readings is shown on fig. 6, in which curve I shows the
relation of the stress to the galvanometer readings. In order to obtain the true reading,
correction must be made for the losses due to radiation and change of resistance. The
(kt\
1 + gj
are shown in the table above, and the plot of these, with the load as abscissee, is shown by
curve II, giving almost exactly a straight line to near the yield point (fig. 6). The stress
strain relation obtained from the extensometer readings is plotted for comparison upon
the same diagram, the unit of extension being 0*00001 inch, and this also exhibits a
nearly linear relation up to the yield point. The result of the experiment appears to
show that the thermal changes do not indicate a range of imperfect elasticity within the
apparent limits of elasticity of shape.
A second experiment upon a wroughtiron specimen having a section 2 inches by
0'25 inches was next subjected to stress in the testing machine in a similar manner, and
the observations are recorded in Table VI., and a plot of the readings is shown in fig. 7.
The observations made to determine the radiation loss are omitted, as they are of a
similar character to the example quoted above. The value of k obtained was 0'0031,
the time being measured in seconds.
The general character of the diagram is the same as in the last case ; there is a
gradual bending over of the galvanometer readings towards the time axis, the deviation
from a straight line being nearly in a geometrical progression with regard to time. The
apparent coincidence of the lower readings with the dotted straight line is probably not
exact. It should be noted that the stressstrain curve would practically coincide with
the corrected thermal stress curve if sheared over, except near the upper end, where the
heating effect begins to play a part. In both cases the thermal readings begin to show
deviations from a linear relation to the stress at about the same value of the stress. In
other experiments upon different bars of iron and steel, results were obtained confirming
those quoted above. It therefore appears probable that the thermal change is very
nearly proportional to the stress, in the same manner as the strain ; and that, for the
material experimented upon, there appears to be no range of imperfect elasticity as
measured by thermal change, coinciding with a part of the range of perfect elasticity as
determined by the strain.
6. The Relation of Stress to Strain and Thermal Change in
Short Compression Members.
It is well known that the relation of stress to strain in short compression members
of wroughtiron and steel follows a linear relation for a considerable range of stress, and
244
PROFESSOR E. G. COKER ON
Table VI.
Time, seconds.
Load, pounds.
Extensometer
Readings.
Observed
Galvanometer
Readings.
Corrected
Galvanometer
Readings.
£=•0031.
1,000
55
10
2,000
65
60
170
173
...
3,000
115
52
330
35
4,000
167
52
520
548
45
5,000
219
53
690
738
55
6,000
272
58
860
938
63
7,000
330
60
1030
1131
75
8,000
390
48
1240
1384
83
9,000
438
59
1390
1569
93
10,000
497
60
1550
1757
103
11,000
557
58
1710
1983
115
12,000
610
60
1860
2191
123
13,000
670
57
1910
2274
132
14,000
727
58
2150
2592
145
15,000
785
63
2300
2817
150
16,000
848
2390
2946
155
17,000
17,500
:
2490
3088
Note. — Distance of scale from galvometer mirror = 6' 0".
that generally there is no very definite yield point — the strain gradually increasing
beyond a certain load, so that the curve showing the relation of stress to strain is well
rounded, and therefore the yield point is not so well defined. In order to obtain pure
compression stress without bending, it is necessary to keep the specimen as short as
possible, and experiments were first made upon compression specimens only long enough
to accommodate the thermopile, the strainmeasuring apparatus being secured to the
compression plates of the testing machine. This arrangement did not give satisfactory
results, and after several trials new specimens were prepared, sufficiently long to allow
of a strainmeasuring instrument being applied to them in addition to the thermopile.
The shortest specimen which could be used under these conditions was 4*5 inches long,
and, as might be expected, the specimen usually failed by buckling, so that it was not
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
245
possible generally to trace the relation of stress to strain and thermal change for pure
compression stress up to the point where the departure from Hooke's law was very
definitely marked ; but sufficient work was accomplished to show that the strain and
thermal change are both proportional to the stress throughout the greater part of the
elastic range, and it seems highly probable, from the evidence obtained in the tension
experiments, that this will hold throughout the whole elastic range of stress, as deter
mined by strain measurements.
In order to indicate the nature of the results the following experiment may be
quoted. The specimen was of wroughtiron, 0*9 x 0'39 inches in section and 4 '5 inches
long. The strainmeasuring instrument used was one specially designed by Professor
Ewing for compression, and similar in principle to the extensometer used in a previous
experiment, except that the distance between the grips was 1*25 inches, and there was
no calibrating screw. The instrument was first calibrated on a Whitworth measuring
machine, and the position of the micrometer eyepiece determined, so that one division
of the scale corresponded exactly to ^roiWo °f an mcn  The instrument was then set
up on the specimen, and the thermopile applied to the broad face. The specimen was,
for convenience, stressed in a small press, provided with an hydraulic diaphragm,
accurately calibrated up to 21,000 pounds, and the load was applied as uniformly as
possible. Preliminary experiments were made to obtain the correction factor for
radiation and conduction, and the value of k was found to be '0096. A load was applied
at the rate of 2000 pounds in ten seconds, until the specimen failed by buckling. The
following readings were obtained in this way : —
Table VII.
Corrected
Time.
Load.
Conipressometer
Reading.
Galvanometer
Reading.
Reading.
£=•0096.
58
10
2,000
58
60
32
3 45
20
4,000
118
59
66
723
30
6,000
177
57
98
1120
40
8,000
234
60
122
1454
50
10,000
294
58
150
1860
60
12,000
352
58
176
2266
70
14,000
Failed by bending
410
196
2618
Notes. — Long connecting leads.
Distance of scale from galvanometer mirror = 10' 2"
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 9).
37
246
PROFESSOR E. G. COKER ON
The specimen failed by bending at the lower end, due no doubt to the compression
plate being provided with a hemispherical seat.
The galvanometer readings uncorrected (fig. 8) show a marked deviation from a
straight line, but the corrected readings follow a linear law almost exactly, and show
no trace of the bending stress, as the thermopile was set against the top of the
specimen farthest from the place of failure.
The strain readings are plotted on the figure for comparison, and it is evident that
the linear correspondence is quite definite. As a further example, the following test
may be quoted of a steel specimen l'Ol x # 38 inches in section and 4f inches in length.
The value of k found by a preliminary test was '0089, and the test gave the following
readings : —
Table VIII.
Extensomoter
Galvanometer
Galvanometer
Readings.
Corrected.
k =0089.
Load, pounds.
Time, seconds.
Reading.
Readings.
Uncorrected.
1,000
57
3,000
10
57
55
28
2 93
5,000
20
112
56
60
652
7,000
295
168
53
93
1050
9,000
40
221
60
115
1350
11,000
51
281
62
139
1700
13,000
60
343
60
166
2100
15,000
70
403
58
185
2430
17,000
81
461
65
201
2730
19,000
90
526
222
3120
100
Went off scale
235
3400
110
Notes. — Long connecting leads.
Distance of scale from galvanometer mirror = 10' 2".
The specimen showed a very slight lateral set near the centre after the application
of the stress, showing that the stress had been carried nearly to the point of failure,
but this is not apparent in the plot of fig. 9.
Other compression tests showed the same general characteristics, and we may
therefore conclude that the thermal change accompanying compression stress for iron
and steel is linearly related to the stress applied through approximately the same
range as the strain.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
247
7. The Variation of Compression Stress in a Long Compression Member.
The stresses in any but a short bar are always influenced by bending, and if we
assume, in pillars of rectangular section, that the lateral deflection is proportional to
the square of the length I, and inversely as the thickness t, it is easy to show, with the
usual assumptions of technical elasticity, that the maximum stress in a long column is
greater than the stress in a short specimen of the same section, by an amount cl 9 /f,
where c is a factor depending upon the fixing of the ends. The value of c for the case
of a pillar with fixed or squared ends is often taken as 30V ¥> where I and t are expressed
in the same units. For technical applications this formula is widely used in the
slightly modified form proposed by Rankine, viz. : —
p
/I
l + el
where p is the allowable working stress,/' is the safe working stress in direct com
pression, and r the least radius of gyration of the section, the constant c being
adjusted to agree as closely as possible with experimental values. As an example of
the ease with which the stresses at different parts of the same column may be compared
by thermal methods, the following result on a specimen with squared ends 15 inches
long and 1*375 by *625 inches in section may be quoted. The load was applied at the
rate of 4000 pounds in 25 seconds, and the galvanometer showed no lag with the rate
of loading. The thermopile was a linear one, and it was pressed against the broad
face of the specimen in the direction of the breadth. The maximum deflections for
different positions of the thermopile were as follows : —
Table IX.
Espt.
1
2
3
4
5
Distance of Pile from
the top end of the
specimen in inches.
Time of appli
cation of Load,
seconds.
Load, pounds.
Maximum
Deflection.
Deflection cor
rected to 70 
4000 lbs.
138
388
750
1125
1338
25
25
25
25
25
704000
704000
704000
703900
1004000
265
300
305
295
290
2'65
300
305
303
290
Notes. — Linear pile, long connecting leads.
Distance of scale from galvanometer mirror = 10' 5".
As the maximum deflections differed very little from one another, and there was no
perceptible lag of the galvanometer, no correction for radiation was necessary. The
actual readings obtained are plotted as ordinates on fig. 10, the distances along the bar
248 PROFESSOR E. G. COKER ON
being used as abscissae, and the corrected maximum readings of column are plotted in
the same way to show the variation of thermal effect.
It will be seen that the corrected readings are not symmetrical with regard to the
centre of the length of the pillar, so that we may infer that the ends were not in exactly
the same condition as regards fixture, and therefore it would be difficult to draw any
definite conclusions ; but the experiments serve to demonstrate the value of the method,
and it appears probable that further experiments in this direction will be fruitful of
results. As a further example of the application of the method, we may quote some
experiments on the variation of thermal effect in beams.
8. The Variation of Stress in the Cross Section of a Beam.
The assumptions of the BernouilliEulerian hypothesis for beams lead to the simple
result that there is a neutral plane perpendicular to the plane of symmetry, and that
the stress at any point of the section varies as the bending moment and as the distance
from the neutral plane. The assumptions of the above theory have been shown to be
false by Pearson,* who has proved that for a beam of circular cross section, subject to
a surface load perpendicular to the axis of the beam, the stress does not vary according
to the distance from a neutral axis, nor according to the bending moment. The varia
tion of stress at the surface of a beam has been determined by more than one
experimenter, chiefly by observations of the strains ; and in order to establish the value
of the thermal method for determinations of this kind, a steel I beam was chosen of the
section shown in fig. 11, and this was subjected to a uniform bending moment by apply
ing equal loads at two points, each distant 4 inches from the central section of a span
of 5 feet. In this way the bending moment at the central section was made as uniform
as possible. The thermopile was pressed against the beam at five different places in
succession, and the deflections of the galvanometer were noted for approximately the
same loading applied at a uniform rate. The value of the correction factor for each
experiment was determined in the usual manner, and its value was found to be very
constant, except in the last set of readings. The observed and corrected readings are
plotted in place upon fig. 11, and from them a curve has been drawn, the ordinates of
which represent to a reduced scale the maximum readings for a total load of 5000
pounds. The variation of thermal change is seen to be proportional to the distance
from a point slightly above the centre line, and (fig. 11) to obey a linear law almost
exactly. These results agree in general with those obtained by Professor Bovey, F.R.S.,t
who used a very delicate roller extensometer. He found an approximately linear relation
for the strains, and in most cases the neutral axis was somewhat above the centre of
gravity of the section towards the compression side, a result which may be expected,
having regard to the probable distortion of the section by the bending moment. The
* " On the Flexure of Heavy Beams subjected to Continuous Systems of Load," Quart. Jour. Math., 1889.
+ " A New Extensometer," Trans. Roy. Soc. Canada, 1901.
THE MEASUREMENT OF STRESS BY THERMAL METHODS.
249
Table X.
Time,
Thermopile on top
of Beam. 4" above
centre. k= 0044,5000
lbs. in 60 seconds.
Thermopile T72
above centre. k =
•0046, 4950 lbs. in
60 seconds.
Thermopile T V'
below centre.
Thermopile 181"
below centre. k =
■0044, 5000 lbs. in
60 seconds.
Thermopile 4" below
centre. £="0054,
5080 lbs. in 60
seconds.
seconds.
GalV.
Reads.
Uorr.
do.
GalV. Corr.
Reads. do.
GalV. Corr.
Reads. do.
GalV. Corr.
Reads. do.
Galv r . Corr.
Reads. do.
10
130
153
•65 66
 85  87
15 T54
20
250
261
175 183
•03
35 370
30
355
378
238 254
•07
2T5 23
58 627
40
490
533
275 30
TO
250 272
65 720
50
580
643
300 335
T2
275 305
7T 806
60
700
792
315 359
14
310 351
77 895
75
760
885
308 361
•14
330 384
77 9T0
90
740
887
290 350
14
325 389
7T 88
105
...
 300  369
Table XL
Distance of pile
from centre line of
beam in inches.
Load, pounds.
Corrected Reading
for Radiation.
Corrected to
5000 lbs.
400
5000
887
887
172
4950
361
363
 063
5000
 T4
 14
181
5000
389
389
400
5080
9T0
909
results obtained were confirmed for the same loading by other tests. In some experi
ments on the variation of stress in cement beams two months old with steel reinforcing,
the thermal method did not give satisfactory results, and apparently the combination
does not appear to behave like a true elastic solid. On the other hand, specimens of
cement of considerable age behave exactly like iron and steel.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 9).
38
250 THE MEASUREMENT OF STRESS BY THERMAL METHODS.
Conclusion.
The application of the thermopile to determine the thermal change in a body, and I
hence the intensity of the stress, has an extremely wide application, since the thermo
elastic equations deduced by Lord Kelvin are generally applicable to elastic bodies
subjected to every type of stress. Only a limited number of cases have been treated in
the preceding pages, chiefly with a view of showing the range of application of the
method. The thermopile, while probably forming the most convenient method of
determining the thermal state of a body under stress, is not the only one which could
be applied. During the summer of 1903, the author, by the kindness of Professor Cox,
Director of the Physics Building, M'Gill University, was enabled to experiment with a
Boys radiomicrometer, which was set up in close proximity to specimens in tension and
compression, and the readings were found to be proportional to the load ; but, with the
limited experience of the author, it was found to be much less convenient than a thermo
pile, mainly because of the extreme delicacy of the apparatus, and the difficulty of
setting it horizontally upon a steady base near the testing machine.
In conclusion, the author desires to express his warmest thanks to Professor Bovey,
F.R.S., for the use of his wellequipped testing laboratory, and to Professor Ruther
ford, F.R.S., for valuable suggestions during the progress of the work; also to Mr
M'Kergow, Demonstrator in Civil Engineering, for his untiring assistance in making
observations.
s R
/. Soc. Edin.
Vol. XLI.
Coker: On the Measurement of Stress by Thermal Methods. — plate i.
oo too no llo , /3c /to ISO t 160 no 180 3&mJ> re FnhZ 1° So a too no mo I3c ;*> lyo /So 170 180 KxhZ.
td&om. '
36o 7Vm£, Seccmds
10 xo 3o q<, so «3o Jo GnLuKH/namdfcr ftcauiZno,
a 2o° foo coo Soo Mo lxo lf „ E^tvnso^luJ BeaM^s
oy. Soc. Ed in.
Coker: On the Measurement of Stress by Thermal Methods. — plate ii.
I jsncie
Vol. XLI.
2?
foo, le
loo, f
v\
f /
J
/ /
s
/ /
' /
/ .
/ /
1
J /
/ / s
/ ' /
i
/ /
/ //
'/ /
/ / /
/ I &al rafiwmetor
jr do
ReacUnqs
Conedvob
JttSfcreessCnxwv
auAvei
O 2o ^O <oo 8o Ti/TLe Sec<7ruis
tyooo 8ooo iZooo ibooo LoixcL fxnjsmdLs
f d s 'o /Z /f /6 &o Hf GoJbttyuTixirn£t£rr Heaaji
loo fco Geo 8oo /coo /too Vruoim, Ext&nsomeitcf
3.0
"— F5 
i
Tv
— ~y
\
^
\
v
A
""*?
h
ao
Fin
» /
/
/
J
;
00
r
il
)
A
"
L 
1
ob O 135 S9fS
r5"
//•2^' /33ff" 15 Bottom;
i
( 251 )
X. — On the Spectrum of Nova Persei and the Structure of its Bands, as photo
graphed at Glasgow. By L. Becker, Ph.D., Professor of Astronomy in the
University of Glasgow. (With Three Plates.)
(MS. received May 2, 1904. Read June 6, 1904. Issued separately September 9, 1904.)
The spectrum of the new star in Perseus, which Dr Anderson, of Edinburgh, discovered
1901 February 21, was photographed at the Glasgow Observatory from 1901 March
3 till 1903 January. From the early photographs one gains the impression that the
spectrum consists of a number of bright bands of different lengths, fading towards the
ends, and overlapping each other, thus producing a series of maxima and minima of
brightness. Near wavelength 5000 the intensity rapidly falls off towards the less
refrangible side, and the bands appear detached. The middle of each of the three most
intense maxima approximately coincide with the hydrogen lines H^ , H y , H s , and on
two photoplates, March 18 to 20 and March 25, each of the bands is crossed by a
sharp Fraunhofer line. On the photoplates taken after 1901 August 1 the bands are
all detached ; some, including the two bands whose middles approximately coincide
with the principal nebular lines, have almost the same lengths, and suggest a line
spectrum in which the lines have been broadened into bands, others are considerably
longer and have pronounced maxima.
While it was probable that the three hydrogen lines and the two principal nebular
lines were represented in the spectrum by bands, it remained to be proved that the
wavelength of a definite point of the band bore a definite relation to the wavelength
of the line to which it belonged. As a result of my investigations, founded on micro
metric measurements and estimates of intensity, I shall show that the bands which con
tain a series of reversals are similar in type, and that the ratio of the distance between
any two points in a band to that between corresponding points in another band is the
ratio between the wavelengths. The spectrum of Nova Aurigae resembled that of
Nova Persei very closely ; its changes followed the same course, and it showed the con
siderable broadening of the lines into bands, the structure of which has, however, never
been investigated. The systematic broadening of the spectral lines into bands, which
for Nova Persei amounted to a 35th of the wavelength in March and April, and a
100th after August, seems to be a feature of new stars, and ought to be accounted for
in an explanation of these objects.
2. The Spectrograph. — The spectrograph of 8 cm. aperture is connected to the
Breadalbane reflector of 51 cm. aperture and 446 cm. focal length. The equatorial
mounting of this instrument, probably made by the late Thomas Grubb some fifty years
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 39
252 PROFESSOR L. BECKER ON
ago, is remarkable in so far as its inclined stand anticipates the chief structural
feature of the Potsdam astrographic refractor. The reflector, which had proved useless
in its old condition, obtained in 1895 a new silveron glass mirror, driving clock with
electric control, driving sector and declination clamp with slow motion, all from the
works of Sir Howard Grubb. The mounting of the spectrograph was supplied by a
local smith. A platform forming a rightangled triangle extends from the upper end of
the tube to the free end of the declination axis, and its plane is inclined 35 degrees to
the optical axis. Parallel to it, the central ray of the reflector is reflected by a plane
mirror. The platform is a stiff structure for its weight. It consists of two layers of
corrugated iron, with the corrugations crossed and bolted at every point of contact, and
it is strengthened by thin sheet steel ribs. To it is clamped a quarterinch steel sole
plate, with adjustable bearings for the two tubes of the spectrograph, and on this
soleplate a small castiron table carrying the prismbox can be adjusted and clamped.
The platform rests at its upper end, a corner of the triangle, on a casting which is bolted
to the tube of the reflector ; at its lower end, the shortest side of the triangle, it is
screwed to a strong castiron arm, which is fixed to the declination axis, in place of the
balancing weights, at right angles to this axis and the axis of the tube. As I had the
declination axis lengthened, and the telescope tube shortened and placed more favourably
in its cradle, the movable part of the instrument weighs now less than in its old condition.
The objectglass of the collimator has an aperture of 8 "2 cm. and focal length of
74 cm. ; that of the camera, a Cooke triplet, 8 '9 cm. by 149 cm. The focal length of
both combined has a large temperature coefficient, 0'13 mm. for a degree centigrade.
The prism made by Hilger of white Jena flint glass measures 16*5 cm. on a side, and is
9 "5 cm. in height. Since it was reannealed its separating power is most satisfactory.
The central portion of the spectrograph is enclosed in a box, and by means of a small
heating apparatus the temperature of the prism and the objectglasses can, at least to
some extent, be kept under control. Unfortunately, the instrument cannot be used in
summer after a sunny clay, because in the iron dome, the large prism is heated in such a
way that the definition becomes too bad for accurate work.
The jaws of the slit are formed by the two halves of a circular mirror 2  5 cm. in
diameter, and they open symmetrically 0*15 mm. for a revolution of the screw. The
width here employed was usually 0'018 mm. The plane of the mirrors which form the
jaws of the slit is inclined 7 degrees to the plane normal to the central ray. If the
image of the star does not fall on the slit, the rays are reflected towards a small mirror
which is fixed to the telescope tube, and thence towards a viewing "telescope" (which
is focussed on the slit) of 7 cm. aperture and 30 cm. focal length (two objectglasses
mounted close together). It lies almost parallel to the collimator. Owing to the large
size of the jaws of the slit, the effective field is half a degree, which is a great con
venience in finding a star and setting it on the slit. The spark apparatus is hinged to
the platform in front of the slit. When turned into position, the optical axis of its lens
coincides with that of the collimator.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 253
The collimator lens, though a fine objectglass, F/9, did not prove to be achro
matised for photographic rays ; the focal curve of the spectrograph runs along zero from
D to Up, then gradually rises, and at wavelength 3700 the ordinate is 2'8 cm. I had
therefore to incline the photographic plate up to 30°.
When the telescope is turned through twentyfour hours of hourangle, the image in
the focal plane of the spectrograph oscillates in simple harmonic motion, with an
amplitude of 0'8 mm., along a line which is slightly inclined to the spectral lines.
1 compensated this deficiency of the mounting by making the plateholder adjustable in
the direction of the spectral lines, and at right angles to them, the motion being effected
by two micrometer screws of '5 mm. pitch. The plateholder takes the plate, 11 by
4 cm., in its upper half, which lies central in the camera, and in the space below it
carries an eyepiece with stout cross wires. As fiducial line for keeping the plate
stationary with regard to the spectrum, I employed the magnesium line 4481, which is
almost covered by the stout wire in the eyepiece. A split spring, which presses against
the jaws of the slit, cuts out a short line from the upper portion of the slit 1"5 cm.
above the optical axis, and the magnesium terminals are placed close to it, inside a short
glass tube, to guard the slit against tarnishing. By these means I am able to keep a
line of the spectrum always on the same place of the photographic plate, and to replace
the plate after days in its old position. The differential change of dispersion due to
changes of temperature is, of course, not taken into account. During an exposure of
the plate I moved the plateholder every time 0*01 revolution of the micrometer screws,
at intervals given by a table, and checked the position once an hour direct on the
magnesium line. To illustrate the efficiency of this method, I mention that on photo
plate No. 23, comparison lines at a distance of 0'04 mm. appear separated, though they
were exposed on twelve different occasions, five seconds each time, on two days, and at
different hourangles.
At the time the new star was announced, wavelength 5200 t.m. was in the centre
of the field of the camera, and 4000 at the end of the plate. No change was made in
the position of the camera until the beginning of October 1901, when wavelength 4170'
was placed, in the centre.
The distance of the hydrogen lines H^ and H v is 20 mm. on the plate, and one tenth
metre or Angstrom unit is represented on the plate by 0"1 mm. at A = 3500, 0"05 mm.
at X = 4300, and 0'025 mm. at X = 5200 t.m.
3. The Measurements and their Reduction to Wave Lengths. — The plates taken in
March and April 1901, and again those after January 1902, were difficult to measure, —
the former, owing to the gradual change of intensity of the spectrum, which presented
few definite points to set on ; the latter, owing to the faintness of the spectrum, some
parts of which could barely be distinguished from the accidental markings on the film.
I finally adopted the rule to measure every point to which the eye was drawn, except
those which I thought to be defects in the film. With respect to these, I became more
careful as the work advanced ; and it is possible that the earlier plates may contain
254 PROFESSOR L. BECKER ON
more than were actually measured of the minima, or reversals, which were present in
the spectrum during the whole period. On the plates taken between August 1901 and
January 1902 the structure of the bands, including the minima, is easily seen, and in
some bands it is visible to the unaided eye. The intensity of the spectrum between
every two points measured was estimated on an arbitrary scale, the estimated " degrees"
of intensity increasing with the intensity. In this paper the " intensity of the
spectrum" stands for the intensity of blackness on the negative, while "intensity of
radiation " is used for the intensity of light in the focal plane of the spectrograph. I
measured each plate about four times, alternately in opposite directions. Each series
includes a number of settings on the lines of the comparison spectrum, ironcalcium
until September, and irontitanium afterwards. The points measured on the same
plate were then identified by a graphical process, and all those were discarded which
had not been repeatedly observed. Only on plate No. 5 I made an exception, where,
after the discussion was finished, I included two minima which had only once been
measured. Since the measuring occupied about half a year, and no measurements were
taken after the reductions were begun, the results of the different plates may be con
sidered independent. For each plate I reduced each series of measurements separately
to wavelengths, and then combined them to mean values. The tables used in the
reductions give the position of the micrometer screw of the measuring apparatus, re
ferred to an arbitrary zero, for each wavelength at an interval of 1 tenthmetre ; they
are based on Ketteler's formula of dispersion,* and were prepared for the angles of in
clination at which the plates were exposed.
The comparison spectrum determines the correction curve of the zero of the table.
The Spectrum in March and April 1901.
4. Results of Measurements. — The results derived from the measurements made on
the photoplates Nos. 1 to 7 are given in Table I. The first column contains the mean
wavelength, the second the average difference of one measurement from the adopted
mean value, and the third, under the heading " Intensity," the estimated degrees of
intensity. The notation ? indicates that the intensity gradually changes from degree
9 to 7.
* Annalen der Physik unci Chemie, 1881, 12.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 255
Table I.
PhotoPlate No. 1, 1901 March 3.
Angle of inclination of plate = 0° ; width of slit 0018 mm. ; orthochromatic plate ; exposure 2*4 hours ;
the spectrum was measured 6 times and referred to 50 standard lines ; good definition of spectrum ; the
more refrangible end beginning at A. = 4500 is increasingly out of focus.
A
V.
Intensity.
A
v.
Intensity.
A
V.
Intensity.
A
V.
Intensity.
3967
5
43177
05
4
48444
07
9
5007
16
3
3972
13
6
43202
06
3
48476
08
•\
5024
6
3
4010
24
43218
04
6
48503
15
= 15
5046
2
1
5
6
03
4049
12
4
43255
03
8
48516
10
►14
= 15
5121
2
1
40778
08
2
43293
06
12
48529
05
= 15
5143
2
4086 2
11
43341
08
48548
12
)
5160
1
5
11
12
3
40905
08
10
43597
07
10
4869
14
8
5210
16
15
40977
04
9
4367
15
9
48732
05
14
5241
3
05
41179
04
9
44128
08
7
48832
07
8
5271
16
1
41237
08
c
7
4435
2
6
48900
08
6
5306
6
3
3
41456
06
O
44542
10
7
48969
05
3
5347
2
3
5
7
25
41613
12
7
4476
*
17
9
4902
1
25
5362
1
15
41865
07
6
4584
3
6
10
4907
13
5
5
5381
2
2
42008
09
5
4619
4
75
49250
07
4
3
5520
2
4
4209
16
4
4640
7
8
49342
5544
9
45
42216
05
c
5
4686
3
7
49431
08
2 ;
5586
1
3
4
42536
10
O
4
5
4
4801
2
5
6
4959
4
1
5678
3
2
4281
5
48352
05
4
4981
2
03
5711
4
1
43125
05
48393
09
3
4985
8
1
5747
3
03
43177
05
48444
07
8
5007
16
2
* The enlargement shows a maximum within this space.
PhotoPlate No. 2, 1901 March 6 and 8.
Angle of inclination of plate = 0°; width of slit 0'018 mm.; orthochromatic plate; exposure l  8
ours, 1 h. on March 6 and 0*8 h. on March 8 ; the plate was intensified and is slightly fogged ; the
spectrum is not so dark as on No. 1 ; it was measured 4 times, 3 times on the intensified plate, and referred
;o 22 standard lines; good definition; focus same as on No. 1.
4089
41182
4296
4319
43246
43286
43378
4347
43555
43592
v. I Intensity.
2
03
2
15
04
04
2
06
04
A
V.
43592
04
43657
07
43756
13
43920
02
44131
04
44660
07
4477
5
4499
4
45280
4545
3
Intensity.
1
2
2
3
35
3
4545
4600
46167
46436
4691
47202
4840
48464
48499
48523
3
18
04
06
2
11
3
07
07
Intensity.
4
3
4
1
2
7
10
48523
48554
48577
48618
4870
48797
48818
48833
4892
4928
09
03
04
2
09
11
01
2
3
Intensity.
10
7
10
8
10
7
05
Table I. is continued at the top of the next four pages.
256
PROFESSOR L. BECKER ON
Table I. — continued.
PhotoPlate No. 3, 1901 March 18 and 20.
Angle of inclination of plate = 13°; width of slit 0018 mm.; Imperial plate; exposure 4*4 hours, T2
h. on March 18 and 3  2 h. on March 20; clear negative; the spectrum was measured 6 times, the Hbands
12 times, and referred to 50 standard lines ; good definition.
A
V.
Intensity.
A
V.
Intensity.
A
v.
Intensity.
A
V.
Intensity.
4028
15
4251
4
7
4512
4
48710
= 9
1
105
13
40348
04
15
4260
4
6
6
4535
2
10
48751
= 9
13
40373
1
9
2
42847
08
7
4551
2
10
48794
11
40440
431531
=
4562
4
11
48839
8
3
7
11
4055
2
43184
\
9
4590
2
10
4895
2
7
3J
6
4061
3
4
(
431903
4602
3
9
4900
3
408076
43196
46072
oi
8
49114
3
4
408151
43269
4611
13
9
4922
11
35
40822
6
43356
12
4620
3
10
10
49312
02
3
40857
7
43409
13
4643
6
11
49410
05
2
40904
4343
14
4679
8
10
4956
2
1
8
14
1
40978
9
43487
14
= 10
4700
3
8
4978
1
15
41027
10
10
43580
12
4735
7
6
5003
4
o
41179
10
43689
10
4794
7
5
5
5025
2
A
15
41211
9
4382
5
9
483346
=
5042
3
1
9
5
05
4132
5
6
6
4396
14
10
48365
)
507
6
02
4148
6
4423
3
10
483743
°
509
10
03
41545
11
5
44278
06
9
48384
!
5133
5
9
8
4165
2
6
4442
2
9
48441
10
5154
02
4187
3
5£
4459
3
105
10
48494
13
5175
02
42039
07
4473
3
48558
5188
05
5
11
9
oi
4219
13
6
4492
3
105
48580
13
5207
oi
4233
16
4502
6
48641
= 9
5265
7
10
13
oi
4251
4
4512
4
48710
= 9
5295
5. The three Hbands. — On photoplates Nos. 3 and 4 three welldefined Fraunhofer
lines are a prominent feature, appearing respectively in the neighbourhood of the
hydrogen lines H^ H y H 5 , and the difference between the wavelength of a Fraunhofer line
and that of a corresponding Hline is proportional to the wavelength. Towards the
less refrangible side of the Fraunhofer lines there are bright bands, which also occur
on the other five photoplates. If the wavelengths of corresponding points of the
three bands be compared with the wavelengths of the three hydrogen lines, the
differences are found also to be proportional to the wavelengths. I conclude that the
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 257
Table I.— continued.
PhotoPlate No. 4, 1901 March 25.
Angle of inclination of plate =13°; width of slit  020 mm.; Imperial plate ; exposure 4 hours ; the
plate is fogged and the spectrum is much fainter than on plate No. 3 ; the spectrum was measured 4 times
and referred to 30 standard lines ; the definition is not so good as on plates Nos. 1, 2, 3, 5.
A
V.
1
Intensity.
A.
V.
Intensity.
A
v.
Intensity.
A
v.
Intensity.
40415
oi
43187
02
\°
4540
4
4
48518
09
02
8
4049
14
05
43200
06
)
35
45597
03
55
3
48580
3
40564
02
1
4325
13
45
4589
13
3
48596
8
40650
05
o
43289
04
55
4613
2
35
48634
07
)
40726
07
3
43352
3
4617
11
6
4
48650
r
40811
04
43378
7
46266
08
8
48661
04
8
40834.
11
4
43449
04
8
46484
07
7
48710
40925
10
55
43540
4660
2
5
6
48728
75
41008
09
6
43546
7
46691
06
6
48799
07
4
4109
16
5
43600
03
6
4682
4
8
4882
1
41156
10
4
43652
03
5
46910
05
7
4884
3
4119
2
3
4372
12
4
4701
3
6
48855
07
2
41220
10
2
4384
17
35
47220
08
4
4894
I
i
15
4135
18
4398
5
3
4729
18
3
49057
06
l
1
2
25
4157
3
44287
25
47399
04
2
4912
12
4165
16
1
4434
13
3
4750
11
15
4929
1
4207
5
2
44546
10
4
4770
4
03
1
4947
11
03
4261
16
44593
07
5
48263
09
4968
15
1
6
2
oi
4277
17
44805
04
48344
11
1
49862
02
15
5
03
42828
08
4493
4
48361
°
5006
6
25
3
05
4307
17
3
45039
08
4
48379
oi
)
3
5020
4
04
43177
02
L
4527
3
45
48462
06
5
50458
08
43187
02
r
4540
4
48518
09
bands are due to hydrogen radiations, which, under ordinary conditions, would produce
the three hydrogen lines. Tables II, III., IV. prove the statement.
Let X be the wavelength of a hydrogen line, A the observed wavelength of a
point in the band, s the correction for the orbital motion of the earth, then
(1) .... X
X + s = A + a.
4500
determines a , belonging to a certain point of a band, and it must be shown that a has
the same value for corresponding points of all three bands. In Table II. the values of
a are compiled. The positions of the first, third, fourth and fifth minima agree as
258
PROFESSOR L. BECKER ON
Table I. — continued.
PhotoPlate No. 5, 1901 March 25.
Angle of inclination of plate = 13° ; width of slit 0*020 mm. ; Imperial plate ; exposure 3 "2 hours ; the
plate is badly fogged ; the spectrum was measured 5 times and referred to 35 standard lines ; good
definition.
40329
40382
40544
40627
4082
4089
40991
4106
4113
41173
41246
4129
4145
41635
4179
42209
42350
03
05
08
10
18
12
04
13
13
04
02
12
17
05
13
oi
08
Intensity.
A
42350
05
4248
1
4279
15
42984
25
4
43138
6
43252 *
7
43269
^
tnin.
43365
)
tnin.
43373
43435
7
6
43459
5
43533
45
43584
4
43620
5
43645
35
43665
4
4377
08
11
3
04
05
05
04
10
03
05
06
Intensity.
5
45
4
5
3
85
6
9
7
10
9
85
85
4
= 4
4377
44055
44272
4454
44713
44773
44863
4507
4556
4582
46057
46154
4660
46978
47247
47538
4806
48255
5
09
06
12
08
08
09
2
4
4
05
06
3
05
08
10
2
07
Intensity.
7
6
7
8
85
9
75
35
3
7
65
55
5
4
A
V.
Intensity.
48255
07
6
48399
02
8
48460
48488
05
11
48558
08
max.
11
48679
08
max.
11
48745
08
9
4878
2
7
4887
14
4
2
48989
10
05
49101
10
9
4932
17
15
49416
07
05
49506
09
1
4963
2
05
5037
16
02
50907
10
* One measurement.
closely as may be expected from the accuracy of the measurements. The sixth
minimum appears as such only in two bands on photoplates Nos. 5 and 6, but each
being measured repeatedly and independently, there can be no doubt that it really
exists ; besides, the other five settings made at this point appear to me to suggest that
it existed also at those places, though it was not appreciated. The second minimum is
questionable, because both positions rest on only one measurement.
The estimated degrees of intensity at corresponding points of the same Hband do
not agree on the seven plates. The observations are too few to determine the
reductions to an average scale of the estimated degrees of intensity as a function of the
degrees ; but since the differences of the estimates made on two plates at corresponding
places are almost the same for all degrees, I reduce the estimates to the scale employed
on photoplate No. 3 by adding to the degrees on plates Nos. 1, 2, 4, 5, 6 respectively
0, 4, 5, 2, 5. The result is given in Table III.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 259
Table I. — continued.
PhotoPlate No. 6, 1901 April 1 and 5.
Angle of inclination of plate = 0°; width of slit O020 mm. ; orthochromatic plate; exposure 6 hours,.
32 h. on April 1 and 2  8 h. on April 5; plate was intensified and the spectrum is faint; the spectrum was
measured 4 times and referred to 30 standard lines; fair definition; focus same as on plate No. 1.
A
v.
Intensity.
A
V.
Intensity.
A
v.
Intensity.
A
V.
Intensity.
4087
1
05
4442
4
3
4731
16
05
4933
13
05
4101
%
08
44622
07
35
476
8
03
4943
3
4114
1
4483
13
3
4800
3
1
5037
6
1?
43107
01
03
4498
3
25
48465
06
5
2
5063
3
43284
08
3
4512
1
*
4853
18
7
5160
2
02?
43375
1fi
=
4
4563
1
48638
06
51770
oi
5
3
4344
18
7
4568
3
35
48651
06
8
5299
6
02
4350
16
=
6
4593
3
q
48707
oi
9
=
538'S
4
05
43559
05
4
4614
2
35
48761
10
5
5447
7
1
43588
09
2
4638
4
4
48789
09
4
5498
7
03
4370
2
t
46631
08
4882
2
3
5578
2
02
35
1
43851
02
05
46781
08
3
4895 4
07
02
5604
3
05
44024
05
2
46900
06
25
4911
4
05
56777
oi
08
44305
04
25
47094
07
1
4925
17
5713
17
02
4442
4
4731
16
4933
13
5739
4
* Defect in film 1
PhotoPlate No. 7, 1901 April 10 to May 3.
Angle of inclination of plate = 0° ; width of slit 0020 mm. ; orthochromatic plate ; exposure, near
horizon, 9 hours on 7 days ; the plate is fogged ; the spectrum was measured twice and is very faint ; bad
definition.
A
Intensity.
A
Intensity.
A
Intensity.
A
Intensity.
A
Intensity.
4325
4360
3
4455
4504
1
3
4614
4718
3
4845
4882
3
5005
5021
1
4536
In Table IV. I have combined the results obtained from three plates to mean values,
in order to exhibit the agreement in position of corresponding points of the three H
bands as reduced to X = 4500, and also to show in what manner the observed degrees
differ. Towards the less refrangible end H Y declines in intensity 4 degrees on 25 tenth
metres and passes into a bright spectrum, while H^ decreases 8 degrees and fades into
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 40
260
PROFESSOR L. BECKER ON
8
■ t^>
s>
c~
8
60
"8
*$
ss
&>
5s
R?
cq
^3
<S
i
f<
w
to
««»
to
1
**o
«>
§s
><$
£
to
§
to
co
o
"8
to
g
St.
8
O 1
tq
5*
A.
i— 3 ^
H *©>
oa *>
*«
^3 "*°
o
J*H =S
o
H '§
o
Gq
<w
^
~<
S
to
^
"8
o
e
Sh
.,
CO
i — i
<w
o
r*
C7>
o
i— 1
O
'■o
^
co
^
SU
to
^3
. 1J
is
8
1
«
■■•££
o
■w
,«
£
*
8
loo
to
m
t~
m
CO
+
IN
OS rC
CM
CN
oo
to
■td
00
rl CO
oo
CO
in
OS
M
as
i>
c3
1
1
+ +
+
+
+
+
+
J^.
t*
*"• V
t^.
CO
CS
CO
5
in
co
M
o
CM
■^ CN
I— 1 )— 1
3
m
CM
CO
<*
o
as
1
1
1 1
+
+
+
+
o
m
CN
m
TlH
tO rt
* OS
■* 11
r^.
OS
tQ.
a
•*
CO
CO
in
CN
CM
IN
■*
OS
CO CN
iH CO
OO o
1— 1
to
CM
00
CN
3
ft
1
1
1
1
1
1 !
+ +
+ +
+
+
f
in
to
to
CO
in
to
to to
o
*K
CN
CN
*
,_,
■a
CO
CO
CN
CM
CN
i— <
CN
to
»— 1
in co
(N
00
CM
1—)
to
o
CN
rH
CO
1—1
1
1
1
1
1
1 I
+
+
+
+
+
+
O
■91
rt
O OS
rl
oo
in
CO
CM
o
CN
1
1
I— I
1
to CO
1 1
o
1
+
to
+
OS
+
oo
CN
+
o
*
o
t^.
OS
CN
CN
to
r^ m
i— I
CN
1
o
CN
1
to
1
CO
to
1
to o
+ +
OS
+
to
CM
+
CN
CO
+
CO
+
O
OS
t»
to
Tl
1
OS
14
+
"3<
rt
to
OS
CO
CO
CO
CO
CO
as
m
OO
OS
<N
OS*
1
1
1
+
+
+
+
+
CM
CO
Tt< to
J> 00
tO iH
CO
o
CO
00
Pt
lO
CO
CN
to ■*
i— ( iH
*tf CO
cn in
CN
00
CN
CO
o
1
1
1 1
1 1
+ +
+
+
+
+
rl
rt
CO
CM
m
O CO
o
in iH
*^
rt
t»
a
g
to
•*
to
CO
1
CN
CO
CN
1
— 1
CM
1
to
1
1
to CO
1 1
+
CO ^
+ +
OS
+
m
cj
+
rl
CO
+
*
CO
CN
to
to
o
to
X
as
M
CO
M
CN
CN
CM
CM
CN
tji
I— 1
IQ
CM
CO
r~
OS
CM
1
1
1
1
1
+
+
+
+
O
H
O
^h
OS
CO
m
CM
1
CO
CN
1
1^
1
CO
1
CO
1
to
_]_
+
00
+
in
CN
to
CO
+
to
CO
J>.
IN
CN
CO
CO
11
OS
CN
1
n
CN
1
i— I
CM
1
to
1
CM
1
OS
+
CN
+
to
i— I
iH
CO
iH
1
+
CM
lO
CO
CO
in
t
t^
0O
as"
in
CO
CN
(N
CO
CO
^
to
CM
OS
CM
1
1
1
1
+
T
+
+
+
t~
#
O
o
in
tH
CO
m
00
m
o
■tn
o
CN
CO
CO
CN
o
CM
o
^
J^
^x
CO
CI
to
CO
a
0)
6
1
1
1
1
1
CO
1
+
+
+
+
+
■"*
os
m
00 o
f~
r^
CO
CO
00
to
CO
in
CO
CN
CN
<N
rH CO
CM rl
n
Tf
o
r~
o
CM
CN
CO
as
13
1
1
1
1
1
1
+
+
+
+
CN
ib
1
+
CO
to
00
as
CM
to
I— I
to
CN
1^
I— I
CN
tl(
t^
CO
CM
1
1
1
1
+
+
•;u
Pd J° "
°N
H
CN
CO
Tt< in co is
00
as o
il CM
CO •*
in to
l^
00
3S
o
CI
CM
"55
a
CO
as
m
CD
a
a
3
a
s
g
a
3
a
1
>>
' HI
a
as
■43
HC
OS
05
cS
CD
o
OS
3
s
cm'
CO
■*
in
to"
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 261
§
rO
"fe.
to
O
8 s
o
«>

1— 1
Sf>
so
<
' CO
CO
to
H
3
<fe
CO
^>
<3C3
■to
* cO
* «o
1
CO
CO
^
O
<M
: cm us
CO
us
F
O OS 00 CO CO
us
us
as
'" H
11
11 V
^H
V
11
11
tS
us
s
US
US
US
00 00 O
o
CM
CO
CO
: co :
CO
CO
i— 1 OS CD 'Hi
CM
CM
CrI
O
c&
W
3
US
V
"■■
W O CO
00
o
CO
CO CO W
CO
us
CO
CO OS 00 t^ CD CD CD
CO
11
11
' l
rH
11
^
X
US
US
US O 00
oo
o
CO
OS CO OS
CO
OS
CO
OS
CO rH OS CI l^
CO
T3
11
'~ l
11
11
rH
tH rH
C
CM
—
»..
■"■ ■»"■ to
CO

rH
: rH :
<M
■^1
tH rH CO US
us
US
us
US
US r* CO CO
00
OS
Ttl
: cm :
~1
00
■^1
rH rH CO CD CO
rH i—t
CM
CM
t^
CO
^.
CO
OS
us o '•
V ^
(N
us
V
^
rH OS t^ CD
US
us
r=
us
us
US »(S
Ph
CO
l~
00
OS
US
o
o
CO rH OS
CI
CM
rH O O O
OS
o
ph
1— <
11
11
rt
r~i tH rH iH
»
^1
rH
US
ITS
US
US
us
us
n
s
t~
1".
00 O 00
u>
OS
o
0O IM
CO
CO
us
<M <M rH O
OS
OS
r\
r— 1
>^
■— 1
rn rH rH rH
[Zi
K
t~
r^
t O OS
OS
,— 1
1—1
• <N TX •
■>JI
o
TjH
rX CSI O
o
OS
T3
C
1 — l
<M
"■
us
io "■ ~;
"~
t^
CM
o cm :
T— I I— 1
M
CO
CO O 00 t^.
CO
i— I
US
US
r« rH CO CO
CO
ou
CM
s
"

rH rH O Ci
OS
OS
CD
us
us
us
co :
CO
CO
US
US
us
US
^
::'
us
us
<D
CO
rH
rH CO
00
CO
OS
: s 1
o
c
'"
O O O) cow
CO
CD
11
11
c
11
O
#
us
• '.°
CD
CD
t^
00 O OS
OS
OS
_
: o :
i— i
5
o
rH
OJ 00 »^ t^
r~
w.
E
3
CO
us
te
CO
*
r* O CO
J~
t.
CO
: os o :
o
o
O O OS OS CO
CO
CO
o
T3
1 — 1
<M
:
CO
: co :
CO
CD
CO
"
r*
r*
rH CM (M
US
US
o
: os
OS
OS
OS OS !>. CO
CO
CD
•imojjo ■<
>JS
i— CCMCOrHUSCOWCOOSOrHCMCOrHUSCOWCOOSO
CI
£
E
S E
E
fl
s
3 3
3
3
E
E
s s
E
3
3 3
3
g
S
§ §
§
r=5
<N
CO' ■"*
US
CO
262
PROFESSOR L. BECKER ON
a faint band. The difference cannot be due to a change in sensitiveness of the photo
graphic film, as its effect is inappreciable within the extent of a band. On the other
hand, it will be proved from the later plates that these bands must be similar also in
intensity, and therefore other radiations must be superposed on the hydrogen
radiations at the ends of the bands.
Table IV.
Table V.
Mean
of the Results from three Plates, Nos. 3, 4, and 5,
showing the Similarity of Structure of the three H bands
and the more rapid Fallin«ofF in Intensity of the HPband.
,c2
Rs
Hy
H>
[ShX
a
Inty.
«o
Inty.
« Inty.
4
7
5
1
43
44
5
7
5
2
32
5
32
8
33
7
3
23
24
24
4*
21
7
 22
9
21
9
5
18
9
9
6
\u
8
16
16
5
2
7
8
14
14
10
11
8
12
12
11
11
13
9
}<
9
 6
 5
8
9
10
 4
 3
10
12
13
11
1 +1
+ 2
+ 1
min.
9
8
12
+ 4
+ 3
10
13
13
13
} +7
1
+ 8
f + 8
= 10
7
14
)
+ 10
10
13
13
15
\
+ 13
1
} +13
min.
fi
} +12
= 9
16
1
J
10
+ 14
12
J
12
17
+ 17
+ 18
+ 16
10
9
115
18
+ 21
8
+ 20
11
+ 21
9
8
19
+ 25
+ 25
+ 26
7
7
10
20
+ 31
6
6
+ 30
95
95
+ 33
6
5
5
21
+ 3G
6
+ 38
9
5
Mean Structure of
Bands.
No.
1
2
3
4
5
6*
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
73
55
43
315
239
218
175
166
144
 120
 58
 38
+ 12
+ 32
+ 68
+ 95
+ 124
+ 130
+ 168
+ 201
+ 251
+ 317
+ 34
+ 364
+ 405
+ 56
Inty.
5
6
7
8
9
3
11
13
7
13
7
125
7
125
7
115
10
55
35
25
1
05
M
?'$ a
S CO 3
PQ
Index
Nos. of
Plates.
3,4
2 I 5
2,3.4,5,6
3,4,5,6
1,3,4,6
3,4,5
The Fraunhofer lines occur only on Plates Nos. 3 and 4, March 18 to 20 and March 25.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 263
6. The Continuous Spectrum. — The question whether the continuous spectrum has
a share in these radiations is settled by photoplate No. 1, which was orthochromatic.
There are five minima recorded between wavelength 496 and 538, whose intensities
ransre between and 0*3 degrees. I obtain the reduction for relative sensitiveness for
different wavelengths from a series of exposures on the sky spectrum. On the
supposition that the intensity curves of the spectrum of the sky and of the star are
identical, the intensity of the continuous spectrum of the star does not reach 1 degree
within the area photographed, and probably is less, considering the observed colour of
the star. Not to complicate the problem unnecessarily, I have disregarded the
continuous spectrum, and it will be seen from what follows that the only effect this
omission can have on the result is to increase slightly the calculated degrees of
intensity of the bands.
7. Mean Structure of the Bands. — The continuous spectrum being very faint, the
difference under discussion must be caused by superposed bands. From the plates
taken after August I shall prove that not only the Hbands, but all the bands, are
similar. I assume that the same holds good for the earlier plates, and that therefore
the superposed bands are similar to the Hbands. The unknown structure of the bands
thus enters not only the Hbands, but also the bands overlapping their ends, and that
structure must be so determined as to give the observed degree of intensity. In the
solution of this problem two other questions are involved : First, in what way does the
intensity curve of the band vary if the intensity of its maximum alters ? Secondly,
what is the resultant intensity on my degree scale if two radiations which singly
produce certain degrees of intensity on the photograph be superposed ?
The two bands, A = 4902 to 4959 and A = 4981 to 5046, agree in extent and in position
of their maxima with the bands calculated from the bands of Table IV. by formula (l),
and the wavelengths of their zero, \, are found respectively to be 4922 and 5016. I
compared the observed degrees of the H^band, which in first approximation served as
standard, with those at corresponding points of the faint bands A =4922 and 5016,
discarding the points where the bands overlap, and deduced by interpolation the
relations given in Table VI. For instance, a band which has at different points the
Table VI. — Corresponding Degrees of Intensity.
13
11
9
7
5
3
1
11
93
75
58
41
23
06
9
75
61
46
31
17
02
7
58
46
3 4
22
10
5
41
31
2*2
13
04
3
23
17
10
04
1
06
02
264 PROFESSOR L. BECKER ON
degrees of intensity 13, 11, 9, 7, etc., has at corresponding points the degrees, say, 5,
41, 31, 22, etc. (Also see § 9.)
The bands contained on the first seven photographs here under discussion give no
evidence as to the second question. On the later plates there are two bands which can
with certainty be identified as consisting each of two bands, while the structure of the
standard band is independently determined from detached bands. From Table XII.,
where the later observations are compiled along with the calculated bands, it will be
seen that the sum of the calculated degrees of intensity due to radiations of the same
wavelength nearly agrees with the observed intensity. I adopt this additive rule here
as a working hypothesis, the accuracy of which will be investigated in § 9.
I deduce the common structure of the bands by successive approximation. Choos
ing first the H^band of Table IV., I calculate the band for A n = 4922, and employing
Table VI., reduce the degrees of intensity so that the degree of intensity of the maximum
of this band agrees with the observed intensity of the maximum. I then subtract
the calculated degrees from their observed values, and find the degrees of intensity at
the different points of the H^band freed at its less refrangible end from the superposed
band 4922. From the corrected Hpband I calculate the band for X = 4265, which a
preliminary discussion had shown to overlap the more refrangible end of the H y band,
and proceeding as before, I obtain the intensities at different points of the more re
frangible end of the H y band. In second approximation I combine these results, unci
repeating the calculation, find the mean structure as contained in Table V., the values
a Q being the means, with regard to weights, of the measurements given in Table
II.
8. Resolving of the Spectrum into Bands. — I set myself the problem to find the
wavelengths X n of the zero of each band, and the degree of intensity of its maximum,
which I shall call the intensity of the band, so that the superposed bands represent the
observed intensity curve. I found this research on the following basis : — 1. The
continuous spectrum is faint and may be neglected. 2. The bands are similar to the
band given in Table V., and determined by formula (l), A being unknown. 3. The
intensity curve of each band is defined by the unknown maximum intensity and the
data contained in Tables V. and VI. 4. At places where bands are superposed, the re
sultant degree of intensity is the sum of the degrees of intensity which the radiations
would singly produce on the photographic plate.
The last assumption is merely a convenient rule, which, though not strictly correct,
is sufficient for our purpose, as will be proved in § 9. I may mention here the con
siderations which induced me to undertake a research which at first sight appears to be
hopeless. I suppose that two bands have been identified in the spectrum, and draw
their intensity curves as calculated from Tables V. and VI., together with the observed
intensity curve of the spectrum. The length, in the direction of the axis of wavelengths,
of the area bordered by the three curves is independent of the manner in which the
ordinates of the two bands are deducted from those of the observed intensity curve.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 265
If this length agrees with the extent of one band, which is almost constant for the
brighter bands, the position of a band can be adjusted in this area, i.e. the wavelength,
X , of the zero of the band can be found. Fainter bands are shorter in length, and two
or more fainter bands may be combined so that their total extent is that of a bright
band. On the understanding that a band selected in the manner described may be
replaced by two or more fainter bands whose values of \ differ little, the wavelength
of the zero of the band is independent of the rule according to which the degrees of
intensity are combined, and depends only on the wavelengths A of the zeros of the two
adjoining bands. The error of the rule will appear only in the residuals. To a lesser
degree, this holds also good for a space belonging to two or more bands, and the greater
the space to be covered, the more important it will be that the intensities be correctly
compounded. For this reason I have divided the spectrum between H 3 and H y into
four parts, introducing three bands, whose maxima I make to coincide with pronounced
maxima of the observed intensity curve. The intermediate spaces were explained by
as few bands as possible, with the intention that each could afterwards be replaced by
two or more fainter bands, should this improve the agreement of the spectrum with one
which otherwise resembled it.
A. For the discussion of the spectrum between H^ and H v I employ the intensity
curves of photoplates Nos. 3 and 4. As before, I add 5 to the degrees of intensity of
No. 4 to refer them to the same scale as was chosen for No. 3. The spectrum on No. 4
differs somewhat from those on the other plates, of which No. 3 is the best representative.
On plate No. 4 there are four prominent maxima between H^ and H v , viz., 4459 to
4480, intensity 6 ; 4560 to 4589, intensity 5'5 ; 4627 to 4648, intensity 8 ; 4682 to
4691, intensity 8 ; all of them fading off several degrees towards both sides. I assume
that they are due to bands whose \ is respectively 4470, 4574, 4637, 4687, and I
calculate the intensity curves of these bands from the data given in Tables V. and VI.
The calculated degrees of intensity were written out at intervals of two tenthmetres,
and subtracted arithmetically from the observed intensities. To explain the residuals, I
chose, in accordance with the above, as few bands as possible, and introduced further the
condition that the same bands be selected for both photoplates. Between 4341 and
4470 at least three bands were required for plate 3 and two for plate 4 ; between 4470
and 4574 three for plate 3 and two for plate 4 ; between 4574 and 4637 one each
for plates 3 and 4 ; between 4637 and 4687 one for plate 3 and none for plate 4 ;
between 4687 and 4861 two bands for both plates. By a lengthy process of trials
in which the wavelengths and the intensities of the bands were altered, including
those of the above bands, I found the wavelengths X of the zeros of the bands and the
degrees of the intensity of their maxima, as given in Table VII. under the heading A.
The three intensity curves calculated from these data at intervals of two tenthmetres
are represented on Plates I. and II. under A, together with the observed curves, which
are dotted. The straight lines drawn at the top of the plates show the extent of each
band and the number of superpositions at each point.
26G PROFESSOR L. BECKER ON
Table VII. further contains the hydrogen spectrum observed by Wilsing and the
spectrum of Nova Aurigee by Vogel, both copied from the table in Wilsing's memoir
Untersuchungen ueber das Spectrum der Nova Aurigce. I ought to mention here
that during the progress of my work I did not consult any previous researches or
observations on new stars, and that I arrived at the result A without bias. Between
4067 and 4341 the wavelength A of the zeros of the bands agree well with the wave
lengths measured in the spectrum of Nova Aurigse, and the two lines 4922 and 5016
are also present in both. On the other hand, there are marked differences between
H^ and H y , the region which, owing to the large interval between two identifiable bands,
presented the greatest difficulty to division into bands. The same remarks apply to the
hydrogen spectrum. It is of no moment that some of the lines, as 4388, 4472, etc., are
found intensely bright, because, in accordance with the above, each might be split into
two or more fainter lines. The fact that the hydrogen line 4581 is the only bright line
of intensity above 6 which is not represented in the spectrum of Nova Persei, while it
occurs in the spectrum of Nova Aurigse, appears to suggest that the observed maximum
4560 to 4589 is not due to the band X = 4570, which is one of those used in subdividing
the spectrum.
B. I therefore repeated the work between H^ and H y , and subdivided the spectrum
as before, but chose 4581 instead of 4570 ; 4634 instead of 4637 ; and 4684 instead of
4687. I further introduced the condition that as few lines as possible should be chosen,
and that where the wavelength of the zero of a band fell near that of a hydrogen line,
the wavelength of the Hline should be taken. The introduction of the line 4581
instead of 4570 as zero of a band changed the position of all the bands as far as the
H r band ; 4570 being the mean of 4559 and 4581, the two bands belonging to them
share almost equally in producing the maximum formerly ascribed to 4570. Each band
entails the introduction of a series of bands fitting into one another, and there are thus
14 bands required to represent the intensity curve compared with 7 bands before. In
other regions I altered some of the wavelengths slightly to make them agree with
those of hydrogen. The band X = 4768, which does not occur in the hydrogen
spectrum, is perhaps due to a series of faint lines. The result of this new analysis is
given in Table VII., B, and the intensity curve calculated at intervals of 2 tenthmetres
is drawn on Plates I. and II. under B. It is remarkable that the lines of the hydrogen
spectrum, which I have been forced to take from Wilsing's table as being within 3 t.m.
of the zeros of bands actually obtained, include all the hydrogen lines whose intensities
exceed 2 between 4341 and 4861, though no heed was taken of their intensity. The
spectrum as defined by A agrees well with that of Nova Aurigse. In the same table I
have further entered all the lines of helium except those of the two second subordinate
series. All of them have corresponding lines in the Nova spectrum.
On the assumption of the additive rule, the brightest lines of the hydrogen and
helium spectrum represent the intensity curve, each line being broadened according
to Table V. and formula 1 ; and I consider the conjecture that hydrogen actually pro
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 267"
duced the spectrum not only possible, but probable, for these reasons: — (l) Towards
the more refrangible side of H Y , lines coinciding within admissible limits with prominent
Hlines, were found without any bias, and they belong to that region of the spectrum
which from the outset offered the least difficulty to the splitting up of the intensity
curve into bands. (2) The intensity curve of the band belonging to 4637 or 4634,
also detected without bias, follows the intensity curve of the March 25 plate in that
part of the spectrum without requiring the introduction of another bright band, and
this line is, together with Hp, H y , H 5 , the most prominent line of the hydrogen spectrum.
(3) If, instead of the line 4570 found under A, which does not occur in the hydrogen
spectrum, the prominent Hline 4581 is introduced, the lines then required to represent
the intensity curve agree in position more closely than before with those of Nova
Aurigse, i.e. a star whose spectrum is, as I shall show, identical with that of Nova Persei
after August. (4) The lines of the hydrogen spectrum, which I have been forced to take
from Wilsing's table as being within 3 t.m. of the zeros of bands actually obtained,
include all the hydrogen lines whose intensities exceed 2 between 4341 and 4861,
though no heed was taken of their intensities.
9. Proof that the Error of the Additive Rule does not affect the Result qf\ 8. — The
results just derived rest on the assumption that the radiations of intensity i x . . . i n which
individually give on the photoplate a blackness of degree m x . . . m n for the same exposure t
produce if acting together during the same time t a blackness equal to 2m. It is, how
ever, well known that this cannot be correct for all degrees of blackness. With the view
of determining the error introduced by the use of the additive rule, I exposed several
plates on a continuous spectrum, each plate containing five spectra, the exposures of
which were proportional to 1, 2, 4, 8, 16. I estimated the degrees of blackness of the
spectra in the same way as done on the star photographs. Any two degrees of blackness
could then be superposed, and compared with those estimated for another exposure. I
find that for the degrees of blackness occurring on the star photographs, the blackness of
the film is about proportional to the time of exposure for a constant intensity of radia
tion, and that the degrees of my scale are about proportional to the blackness. Since
this relation cannot hold good for the highest degrees of blackness, I take it to be only
approximately true for the lowest degrees, and put
(2) 1 = Q. Lj for a radiation of constant intensity i,
k fi?>h)
and choose
(3) /(m) = 10 om l,
where m is the degree of blackness, t the time, and a a constant.
Scheiner's Die Photographie der Gestirne contains on p. 246 a table, the results
of experiments by Michalke, which gives the times of exposure for intensities of radia
tion varying from 1 to 36, to produce the same degree of blackness on the photographic
film. I find this table is sufficiently well represented by
(4) ti b = constant 6 = l  08, for a constant blackness m.
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 10). 41
268 PROFESSOR L. BECKER ON
Table VII. — Spectrum of Nova Persei in March and April 1901.
Nova Persei.
Hydrogen H
(Wilsing).
Helium.
Nova Auiigae *
(Vogel).
A
H.
Intensity
Intensity
r t
o
o
CN
O)
V
CO
S
id
CN
A .
CO
03
1
CO
o
1
CO
^1
A.
]
A.
I.
Series.
A.
I.
c3
a
3
a
c8
a
1
o
a
ea
a
4026
5
l
5
4026
5
l
5
4027
4044
4055
4
3
3
3
2
4026
5
11,1
( 4063
6
6
4067
i
2 5
4067
25
• 4067
 4070
4088
4097
5
5
3
2
5
5
1
1
4067
b 3
4102
9
10
10
4102
9
10
10
4101
5
1
4102
b 3
4122
2
2
2
41 32
3
3
3
4132
4
2
4125
2
4145
2
2
2
4145
1
1
4145
3
1
4144
2
II, 1
4160
15
15
15
f 4157
1 4163
4171
5
5
4
2
2
3
4158
2
4174
6
6
6
4177
55
45
45
4177
4182
4189
4196
8
3
2
3
8
1
3
4176
vb 3
4210
2
2
3
4208
15
15
15
j 4205
 4213
4222
6
7
4
4
5
3
4234
45
45
45
4233
45
5
5
4233
4243
4253
3
1
2
1
1
1
4230
b 3
4265
35
45
45
4265
35
45
45
4265
4293
1
1
4262
4288
vb 3
b 3
4304
1
3
3
4305
1
3
3
4305
4312
4330
3
2
2
3
4315
b 3
4341
9
105
105
4341
4364
4377
9
2
1
105
3
105
2
2
4341
4377
4382
13
2
3
1
4341
4383
b 4
3
4388
85
85
■ 85
4388
55
45
4
4389
4402
2
2
1
1
4388
3
II, 1
4410
2
4410
4
5
3
4410
4413
4419
4
2
2
3
4417
b 3
4436
5
5
3 5
J 4425
\ 4448
1
25
2
4425
4
4
4435
b 3
35
3
3
4448
2
4
4445
2
4459
■ • •
3
35
4459
3
4
4472
4
85
9
4472
3
4
3
4472
6
1,1
4473
b 2
I. = intensity.
I = helium, II = parhelium, p = principal series, 1= first subordinate series.
In the last column b = broad, 3 = bright, 4 = very bright.
* Copied from Publicutionen des Adrophysikalisclien Observatoviums zu Potsdam, xii. p. 96.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 269
Table VII. — continued.
Nova Persei.
Hydrogen
(Wilsing).
Nova Aurigee
(Vogel).
A.
B
Helium.
Intensity.
Intensity.
©
o
I>5
A .
CO
V
CO
03
id
p
00
1 — 1
a
o
,4
o
ti
00
*P
A.
]
[.
A.
I.
Series.
A.
I.
3
p
3
3
3
3
3
4482
1
1
4480
)
4488
3
3
4
4488
2
4
Us
4502
5
3
4500
3
4
1
J 4498
1 4501
2
2
4
4495
4507
1
2
4520
5
6
6
4522
25
3
5
1 4521
 4523
2
1
2
3
4520
lb 3
4535
4
3
2
4535
4551
4555
2
2
2
2
2
2
4530
f
4558
4
25
2
4559
6
5
4
4559
5
2
4557
b 2
4570
7
8
8
4581
3
55
5
4581
4597
6
2
10
1
4583
b 3
4612
35
5
3
4608
2
4
1
4608
4619
2
2
1
1
4637
35
35
8
4634
5
6
9
4634
9
15
4628
b 3
4660
3
6
4662
4672
25
25
3
4662
4672
1
1
2
3
4687
5
55
10
4684
5
6
10
4684
4710
4719
3
2
2
3
2
2
4725
45
45
5
4724
4
4
6
4724
o
3
4768
55
55
55
4768
55
55
55
4797
1
4861
13
13
13
4861
13
13
13
4862
7
2
4862
b 3
4922
4
4
4
4922
4
4
4
4932
4973
2
4922
4
II, 1
4924
2
5016
35
05
2
5016
3
05
2
5014
5055
5016
6
II, p
5016
o
5132
1
5132
1
5178
2
5178
2
5167
vb 3
5200
2
5200
2
523
528
2
2
5327
2
5327
5405
3
2
5388
5317
4
5456
3
5451
3
5481
5500
25
5495
25
5499
5505
5544
4
5537
4
5537
5589
3
5584
3
5635
25
5640
3
5680
2
5689
2
5689
'
The hydrogen lines 4973 to 5689 were measured by Hasselberg.
270
PROFESSOR L. BECKER ON
On p. 247 Prof. Scheiner further gives the results of his experiments on artificial
stars produced by means of a Zoellner photometer. He finds that if the time of
exposure be increased 2'5 times, the faintest stars recorded on the plate are only 0'7 mg.
fainter than before, therefore b = 1*4.
Let a radiation of intensity i produce a degree of blackness m, and a radiation of
intensity i' a blackness m', both in time t. Let t! be the time required for i to produce
m', then by (2) and (4)
ti' b = t'i" and , = $2?). therefore
t f(m )
f(m) f i\''
(5) v / = . for the same exposure on radiations of intensities i and i .
■ ' /(to') \i'J *
I assume that the broadening of the lines into bands is due to the same physical
cause, and that the ratio of the intensities of the radiations at any two corresponding
points of two bands is a constant for these two bands ; therefore by (5)
(6)
^" = constant, and ^7^/ = Jr~7\ = constant,
/(to)
fim') /(//) /(/,)  /(,*')
where n and m are the degrees of blackness at two points of a band, and 11! and m' those
at corresponding points of a second band. By means of (6) I determine the constant
a in f(m) from the observed corresponding degrees of blackness contained in Table VI.
The result is a = 0'04, with which I have calculated the following table.
Table VIII.
Calculated corresponding degrees of blackness for a = O04.
H
13
11
9
7
5
3
1
11
92
74
57
40
23
07
9
74
59
45
31
18
05
7
B7
45
34
22
12
04
m 
5
40
31
22
15
08
02
3
23
18
12
08
04
oi
1
07
05
04
02
oi
For instance, if the maximum of a band of degree 13 is reduced to degree 5 in
another band, blacknesses 7 and 5 at other points of the band become respectively 2 '2
and 1*5, while Table VI. gives 2 '2 and 1*3. The quantities in this table differ from those
in Table VI. for all degrees of blackness greater than 0*8 by less than 0"2 degrees, and the
average error is 0"1, but the differences increase to 0*4 for the degrees lower than 0'8.
The function (3) therefore represents the observations satisfactorily. For a = 0*03 and
0*05 the residuals are respectively 30 and 50 per cent, higher than for a = 0'04, and for
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 271
<j =  07 they increase to 2^ times the amount for a = 0'04. For double the exposure
the degrees of blackness 1, 3, 5 and 10 become respectively 1*9, 5'3, 8*4 and 15.
Let the intensities of radiation i x . . . i n give respectively the blacknesses of degree
<m x . . . m n when exposed time t. The question is, what is the degree m if a
radiation of intensity 2{ be exposed the same time t ? Formula (5) gives
i B F(m ) = F(ra„)2i where F(ra) = (/(m))» , therefore
(7) F(m ) = 2F(ro M ).
An extract of the FTable is given below.
Table IX.
m.
F(m).
m.
F(
n).
6=108.
6=14.
6=108.
6 = 14.
1
2
3
4
5
6
7
00
oii
023
035
047
061
075
091
ooo
019
032
044
056
068
081
093
8
9
10
11
12
13
14
108
127
147
168
192
217
245
106
120
134
149
165
182
200
For instance, radiations which would singly give the degrees 1, 2, 3, 4 produce when
superposed m =8'4 for 6 = 108 (by Table IX., F(m ) = 011 +0'23 + 0*35 + 017 = 116)
and 11*1 for b = 1'4, as compared with 2m =10.
The difference ?n — 2m depends on 2m and the number of radiations which are
superposed. It varies most for the intensity curve which shows the greatest range
between maxima and minima, and which has a different number of superposed bands at
different places. On this account the calculated intensity curve, 2m, belonging to photo
plate No. 4, which on hypothesis B has from 3 to 6 superposed radiations in the region
between H^ and H y is the most likely to differ from an intensity curve m calculated by
formula (7). I assume the same bands, and in first approximation the same degrees m
which contribute to the 2mcurve of Plate I., but I compound them according to formula
(7). The resulting ?w curve is then brought to agreement with the observed intensity
curve by suitable changes of the intensities of the bands, and the question is whether
this curve satisfies the observed curve as well as the 2m curve does. Table X. contains
the calculation in detail at an interval of 10 t.m., while it was made for every 5 t.m.
The first columns show, under the heading m, the degrees of blackness at each point which
the radiations would singly produce on the film. 2m is the ordinate of the calculated
intensity curve on Plate I., which was made to agree as near as possible with the
observed intensity curve (see § 8) ; m is the resultant if the degrees m be compounded
272 PROFESSOR L. BECKER ON
according to formula (7). m 2m ranges between — 0*6 and —2*2 for 6 = 1*08, and
between +0*4 and + 2*4 for 6 = 1*4.
Let Smx . . . Sm n be the corrections of the degrees m x . . . m n which change the
degree of blackness m by Sm therefore,
(8) S^Ogm,, = dF ( m °hm .
dm n dm
A change Sm^ at wavelength X of band 1 can be brought about only by all the
degrees of blackness being changed at every place of this band. I express Sm 1 by the
change Snj of the degree of blackness /*i of the maximum of band 1 ; m x being the
quantity which was shortly called the intensity of the band, and I do the same for all the
indices 1 to n. Employing (6) I replace in (8) $m n by <V» and eliminate function /by F.
(9) 2F(m/ lo g/fc"V = dF }™°hm .
dfi n dm^
To reduce the work of calculating, I change the degrees of the maximum of every
band by an amount <V n determined by
(io) g= di °g/foy,
dfx n
and obtain from (9) and (7)
(11) ^logFK),
dm
(m ) = m + 8m .
I determine x by (ll) with Sm = ^m — m , and calculate $m at each point
from the mean value of x. For 6=1'4, //. of band \ = 4425 was reduced by 1 in
addition to <V, and the intensities of the last four bands were not changed at all.
(m ) = m + $m which appears in Table X. is then the calculated intensity curve if the
degrees of blackness of the maxima of the bands given in Table VII. be changed by certain
amounts to be calculated from (10), and the degrees at each point be compounded ac
cording to (7). The last three columns give the residuals left in the observed intensity
curve. 2m differs from the observed intensity curve on an average 0*58 degree, (m )
differs 0'57 for b= T08, and 0'63 for 6= 1*4, while (m ) differs on an average 0*26 from
2m. The observed intensity curve is therefore equally well represented by 2m and by
(m ), and therefore the hydrogen lines of Table VII. represent the spectrum, no matter
whether the degrees of intensity of their bands be compounded by mere addition, or
according to formula (7). Combinations of bands which, if compounded according to
the additive rule, leave inadmissible residuals in the observed intensity curve, must give
errors of the same order if formula (7) be employed ; and I conclude that if this formula
had been used at the outset in analysing the spectrum into bands, the result would have
been identical with that contained in Table VII.
A similar calculation for the intensity curve of photoplate No. 3, assumption A,
gave an average error, observed m — 2m = 0*63, observed m — (m ) = 0'59 for 6=1*08,
and 0'58 for 6 = 1*4, while (m ) differs on an average 0'25 from 2m for both values of
6. This agreement proves again that the use of the additive rule cannot have influenced
the analysing of the spectrum into bands.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 273
Table X.
Calculated Intensity Curves, 2m and (m ),for Photographic Plate
No. 4.
O M
T2
4360
70
80
90
4400
10
20
30
40
50
60
70
80
90
4500
10
20
30
40
50
60
70
80
90
4600
10
20
30
40
50
60
70 J
80
90
4700
12
95
9
82
7
7
7
75
10
11
10
10
8
9
9
95
95
85
8
105
105
11
13
13
12
105
11
11
13
12
Intensity Curves of the different
Superposed Bands.
65
42
02
04
06
ro
14
30
30
28
11
06
08
11
16
21
40
40
38
21
ro
04
24
36
44
48
60
84
10
9 6
20
19
12
04
0'9
20
19
16
04
oi
10
10
09
02
02
07
21
19
16
05
04
05
06
l'l
16
22
40
38
36
21
10
02
10
09
08
08
17
22
25
34
16
20
40
38
38
19
04
08
1"2
23
30
28
IT.
0'5
04
13
17
2 2
27
50
50
48
27
15
04
06
1
14
30
3
28
1 5
05
04
13
17
22
27
41
50
48
38
15
06
8
13
17
35
35
33
15
01
0'4
09
20
T9
16
04
112
108
81
70
8i
70
70
70
82
92
106
109
101
85
85
83
86
9'7
95
9::
71
107
103
99
76
81
107
126
130
121
103
114
122
121
124
by Formulas (7) and (11).
6 = T0?.
94
89
72
64
72
62
63
62
70
78
87
89
84
74
7 3
72
76
84
80
79
65
90
88
85
68
72
95
110
111
104
92
101
112
110
109
i
(m ).
6=14.
+ 18
108
+ 19
102
+ 09
83
+ 06
74
+ 09
83
+ 08
72
+ 07
73
+ 08
72
+ 12
81
+ 14
90
+ 19
100
+ 20
102
+ 17
97
+ 11
86
+ 12
83
+ 11
83
+ 10
88
+ 13
97
+ T5
92
+ 14
91
+ 06
75
+ 17
103
+ 15
101
+ 14
98
+ 08
79
+ 09
83
+ 12
108
+ 16
125
+ 19
126
+ 17
119
+ 11
105
+ 13
115
+ 10
127
+ M
123
+ 15
124
122
122
96
84
96
88
90
94
104
108
120
122
114
100
104
100
104
110
108
106
81
112
110
108
86
94
112
124
126
118
110
118
122
120
124
W
10
14
15
14
15
18
20
24
22
16
14
13
13
15
19
17
18
13
13
13
10
05
07
09
10
13
05
+ 02
+ 04
+ 03
07
04
oo
+ 01
oo
(m ).
107
107
83
72
80
70
67
71
82
91
105
107
99
86
90
86
90
96
94
92
69
100
100
103
81
91
112
124
126
118
110
118
122
120
124
Kesiduals
I
O
+ 08
13
+ 09
+ 12
11
oo
oo
+ 05
02
12
06
+ 01
oi
+ 15
05
+ 07
+ 04
02
oo
08
+ 09
02
+ 02
19
+ 04
01
+ 03
+ 04
oo
01
+ 02
04
12
+ 09
04
 (m ).
+ 12
07
+ 07
+ 08
13
02
03
+ 03
oi
10
oo
+ 08
+ 03
+ 14
03
+ 07
+ 02
02
+ 03
06
+ 05
+ 02
+ 04
18
+ 01
03
+ 02
+ 05
+ 04
+ 01
oo
05
17
+ 07
04
+ 13
12
+ 07 I
+ 10
10
oo
+ 03
+ 04
02
11
05
+ 03
+ 01
+ 14
10
+ 04
oo
oi
+ 01
07
+ 11
+ 05
405
23
oi
11
 02
+ 06
+ 04
+ 0'2
05
08
12
+ 10
04
274 PROFESSOR L. BECKER ON
The Spectrum from 1901 August 1 to 1902 November.
10. The Mean Spectrum. — The results derived from the photoplates Nos. 8 to 21,.
1901 August 1 to 1902 January, agree closely with each other, and it is unnecessary
that the results be given separately. Those derived from the later plates, Nos. 22 to
27, are considerably less accurate, owing to the faintness of the spectrum, but they suffice
to show that the bands did not change in position. Their results are also not printed
separately. The changes which the spectrum underwent belong to the intensity, and
they appear for the whole period in Table XVII.
I combined the wavelengths and the estimates of intensity to mean values, which
are given in the first columns of Table XII. For most bands seven to eight plates
contributed to the mean, and for the band near H y thirteen plates, all belonging to the
period August to November. The two bands at wavelengths 386 and 397 were outside
the range of the plate until the beginning of October, and they rest on the results of
the plates Nos. 18, 19, 21. In Table XII. the intensities of the bands, therefore, do not
belong to the same epoch. The average error of a tabulated wavelength is 0*3 t.m.
With reference to the faint bands, whose intensities do not exceed 1*5, and which
were difficult to measure, most of the detail had to be discarded, because it was seen only
on one plate. The neglected measurements are about five per cent, of the total number.
The wavelengths of these faint bands may be several tenthmetres wrong.
11. The Common Structure of the Bands. — Of the detached bands, the first two
have the most pronounced intensity curve (see Plate III.). I shall show (see § 12) that
their structure is similar, and further, that the similarity extends to all the other bands.
The wavelengths of the lines to which the bands belong being unknown, I introduce
\ n , the mean of the wavelengths of the three minima, and determine a m from
(12) ^.. + «iW0'
a m is given in Table XIII. for the first two bands, and two other prominent bands, whose
zeros X TO equal 4364 and 4726. The latter merge into fainter bands, which overlap their
more refrangible ends ; the intensities at these places are bracketed in the table, and I
do not take them into account here. The adopted values of Table XIV. are the means of
the figures contained in Table XIII. , with the exception of a few which were corrected so
as to represent other bands better. I have also calculated a m for all the other bands
except one, and drawn on Plate III. their intensity curves with a m as abscissa.
12. The Calculated Spectrum. — I decide whether all the brighter bands are of the
same type by calculating the different points of the bands from a m of Table XIV. by means
of formula (12). About six welldefined points contributed to the final X m , and its
calculated error is on an average 0"2 t.m. The degrees of intensity have been obtained
from Tables VIII. and XIV. I have given the calculated bands in Table XII. Including
all points, I find that the calculated wavelengths differ from the observed wavelengths
on an average 0'5 t.m., as compared with a calculated average error of 0'3 t.m. for A, of
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 275
Table XI
Photog
raphic j
Plates,
1901
August to 1903 Februai
T
<D
Number of
+3
c3
Points Measured
S
5
Z3
60 r<
Definition
o
a<X
A
c3 '~ H
en
« 53
Date.
«
Remarks.
C4H
43
M J>
H
O
a

Star.
Fe, Ti.
S
<D
P
Q
3
fe
Hours.
Degrees
Centigrade
Star.
Fe.
B
s
8
1901. Aug. 1
17
05
4
3
Intensified
35
25
4
6
9
Aug. 1 5
40
20
2
2
Intensified
93
24
15
8
10
Aug. 21
65
29
2
2
98
32
14
6
11
Aug. 26
58
18
2
2
109
40
18
4
12
Aug. 27
59
15
3
3
Intensified
77
24
10
3
13
Aug. 30
4 3
18
2
2
Intensified
94
25
14
4
14
Sept. 4
77
19
2
2
Very clear
114
33
20
4
15
Sept. 20
47
02
4F
3
Intensified
49
17
5
2
16
Oct. 2
Oct. 4
37
19
30
4F
3
Intensified
45
25
2
2
17
Oct. 6
70
13
2
2
Intensified
108
40
15
4
18
Oct. 31
85
12
2
2
64
30
12
4
19
Nov. 1
60
02
2
2
77
36
12
4
20
1901. Nov. 13
55
05
2
2
Intensified
133
45
17
4
21
1902. Jan. 12
58
11
2
1
Intensified
87
50
5
4
22
Jan. 26
Jan. 28
16
63
07
F
2
Intensified
77
45
2
4
23
Jan. 29
Jan. 31
76
76
22
F
1
Intensified
80
35
1
4
24
Feb. 9
Feb. 11
Feb. 12
Mar. 21
48
48
43
20
20
F
2
Intensified
81
42
4
4
25
April 1
April 2
April 17
April 26
April 30
May 1
May 2
26
18
15
15
08
09
14
50
vF
1
Intensified
78
52
2
4
26
Oct. 20
Oct. 26
Oct. 30
Nov. 1
22
90
33
85
33
F
2
Intensified
98
48
3
4
27
Nov. 17
Nov. 18
Nov. 20
1902. Nov. 21
1903. Jan. 7
Jan. 24
Feb. 1
67
67
69
22
05
05
05
17
vF
1
Intensified
46
32
2
1
4
"Width of slit:— 0018 mm. for plates Nos. 8 to 25; 0020 mm. for plate No. 26; and 0022 mm. for
No. 27.
Angle of inclination of plate: 13° for Nos. 8 to 17 ; 16° for Nos. 20, 22, 23, 25; 30° for Nos. 18, 19,
21, 24, 26; 8° for No. 27.
Definition:— 1 excellent, 4 inferior, F faint.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 42
276 PROFESSOR L. BECKER ON
Table XII.
The Observed Spectrum, 1901 August 1 to 1902 January, and the Calculated Bands.
Observed.
Calculated.
Observ
ed.
Calculated.
No. of
No. of
A
Intensity.
point
Table XIV.
A
Intensity.
A
Intensity.
point
Table XIV.
A
Intensity.
3813
05
A
,,, = 38695
A
M = 39680
3835
38535
38560
3
1
o
536
563
21
39785
39807
6
o
14 1
15/
16
780
803
62
38581
4
3
581
32
39823
£1
1
17
819
26
7
95
3986
09
38594
11
4
596
115
4027
18
858
38611
8
5
614
95
4045
1
38626
1
6
627
11
4063
38638
3
7
639
21
40715
05
38652
11
8
652
105
40797
1
38679
9
683
o
Kn >
6
53
40826
38708
10
705
10
105
2
1
861
38729
11
728
04
9
95
40883
2
894
38741
12
741
15
07
38765
5
13
764
32
40911
3
913
26
38789
10
8
14)
15/
792
84
74
3
4
5
929
948
34
38814
16
815
26
3883
3
17
831
32
40958
6
962
02
1
11
1
7
975
3889
18
869
04
3900
03
40981
25
8
988
3
39366
>
.„, = 3968(
)
41028
07
9
021
12
39432
05
41053
10
045
30
395181
1
1
513
25
11
069
2 6
0?
17
41092
12
083
39543
2
544
05
07
39557
2
3
563
26
41104
13
14 (
15 j
107
22
39575
4
4
579
80
25
137
19
39594
10
5
597
99
41153
16
162
0'7
39614
7
6
611
80
15
17
178
02
09
41213
18
218
2
7
622
39639
8
636
17
4140 ±
 5
39665
10
9
668
90
4165 ±
o
39693
4
10
690
44
4220 ±
05
9
11
713
90
4265 ±
39729
12
728
80
4300 ±
03
39744
3
13
750
26
4306
07
7
71
43142
1
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 277
Table XII. — continued.
Observed.
Calculated.
No. of
No. of
No. of
A
Intensity.
Point
Table XIV.
\
Intensity.
Point
Table XIV.
\
Intensity.
Point
Table XIV
A
Intensity.
X
„ = 43410
43232
17
1
228
06
43266
2
2
261
10
43288
4
3
4
5
282
29D
319
36
45
36
43332
15
6
7
335
347
03
06
43359
4
8
362
40
43401
9
396
17
43420
45
10
11
422
447
40
36
\ m = 43645
43463
1
12
462
10
1
462
20
43474
6
13
487
43496
9
14 )
30
2
496
30
43517
15/
519
3
516
90
4
53 3
12
16
545
26
110
43554
7
17
562
10
5
554
90
43574
1
03
6
7
569
582
10
20
43590
10
8
596
100
43614
9
18
605
43633
9
631
5
50
43653
10
10
657
100
43679
8
11
682
90
43698
3
12
697
30
43718
8
5
13
723
8
5
43745
14
749
43757
15
761
7
7
43784
3
16
17
781
798
3
1
43820
1
■ 18
841
27S
PROFESSOR L. BECKER ON
Table XII. — continued.
Observed.
Calculated.
No. of
No. of
No. of
A
Intensity.
Point
Table XIV
A
Intensity.
Point
Table XIV
A
Intensity.
Point
Table XIV
A
Intensity.
43887
08
43933
02
4398
4405
03
44467
1
44574
15
4488
4503
15
4554
4570
i
15
1
4578 3
K=4
16126
4590
1
932
15
2
968
02
03
45988
3
990
13
4
008
2
5
030
17
13
46044
15
6
7
8
046
059
075
01
02
15
46110
9
111
(>
.,,, = 46353)
1
06
46145
10
139
15
15
11
165
13
1
(158)
46193
, 12
1 13
181
03
K= 46420
08
208
2
(194)
12
2
3
(216)
43
46233
14,
15 J
11
1
225
4
(234)
242
08
56
25
16
269
09
2
261
12
5
(256)
43
46277
03
3
283
6
(273)
04
17
288
43
7
(286)
08
5
01
4
301
56
8
(301)
18
333
5
32*3
43
50
46340
6
340
9
(339)
2
04
22
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 279
Table XII. — continued.
Observed.
Calculated.
, No. of
No. of
No. of
A
Intensity
. Point
Table XIV
A
Intensity
Point
Table XIV
A
Intensity
Point
Table XIV
A
Intensity.
7
353
08
46364
5
1 8
368
50
10
11
(365)
(392)
50
43
46411
3
9
406
22
12
(409)
1"2
46435
5
10
432
50
! 13
(436)
46453
14 ,
15 J
38
11
459
(470)
,45
43
33
I 19
1 13
476
46489
503
12
16
(497)
4
,14
I 15
38
12
46520
537
17
(516)
35
33
04
46550
3
16
564
12
18
(561)
46584
1
X
m = 46878
17
18
583
628
04
46691
3
1
2
681
718
09
14
46736
3
739
5
4
5
758
780
48
62
48
46788
1
6
7
797
810
5
09
46811
6
8
\ 9
I 10
826
863
55
4687
25
891
5
55
46914
11
918
46942
4
2
12
934
48
14
1
,,= 47156
958
46972
5
13
961
42
09
46997
7
14,
15 i
16
996
024
36
14
2
3
4
995
017
035
14
48
47040
6
17
043
05
5
057
62
48
K
,= 47261
47074
25
18
088
_
6
7
074
088
05
09
l
063
14
280
PROFESSOR L. BECKER ON
Table XII. — continued.
Observed.
Calculated.
No. of
No. of
No. of
A.
Intensity.
Point
Table XIV.
A
Intensity.
Point
Table XIV.
A
Intensity.
Point
Table XIV.
A
Intensity.
A m = 47156
A
,, = 47261
47102
9
8
104
55
2
099
22
47125
11
9
10
141
169
25
55
3
4
5
121
140
162
71
88
71
47172
7
11
196
48
6
7
179
193
07
14
47210
10
12
213
14
8
209
80
47250
7
13
"1
15 J
240
42
9
246
38
47270
9
274
3 6
10
274
80
16
303
11
301
14
71
47317
2
17
322
05
12
318
22
47354
18
368
13
15 i
345
62
55
380
54
47403
3
16
408
22
47426
1
17
427
07
47473
05
18
473
4757
03
4768
4776
05
4786
4799
05
4810
4824
05
48342
48403
1
A
m = 48628
48425
1
1
424
07
48454
27
2
462
11
48483
5
3
4
484
504
40
50
48519
45
5
526
40
48538
1
6
7
544
558
04
07
48569
4
8
574
45
48617
15
9
613
20
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 281
Table XII. — continued.
Observed.
Calculated.
Observed.
1
Calculated.
No. of
No. of
A.
Intensity.
Point
Table XIV.
A.
Intensity.
\
Intensity.
Point
Table XIV.
A.
Intensity.
48642
10
641
A m = 49594
40
45
14 ,
15 J
48666
35
11
669
40
719
I 13
686
12
48700
714
11
4973
08
16
748
04
28
34
4978
17
768
48754
}
15 J
750
18
817
01
15
29
02
48787
16
779
1
11
K
17
799
48831
18
846
04
49862
05
1
862
07
4886
9
901
11
4900
05
03
49921
3
924
40
4917
4924
;
l m = 4959
1
5
4
5
944
967
5
40
4938
05
1
l
2
386
424
03
49983
05
6
7
986
ooo
04
07
49448
3
447
04
50010
4
8
016
45
4
467
17
50065
9
056
20
25
5
490
23
50099
10
085
45
49505
6
508
17
3
11
114
40
04
7
523
oi
50175
1 13
132
11
49542
8
539
03
161
2
9
10
579
607
20
08
28
15 J
198
34
27
49640
11
636
20
50220
17
16
228
11
1
12
654
17
50261
07
17
248
04
13
682
04
5031
03
18
297
15
5053
282
PROFESSOR L. BECKER ON
Table XIII. — Table of a m of Corresponding Points of Four
Bands and of the Intensities, showing that the Bands are
similar in regard to their Wave Lengths after August 1,
1901.
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
a in Tenthmetres.
38694
185
156
131
116
 96
 79
 65
 49
 17
16
41
55
83
39680 43644 47263
155
139
119
 98
 75
 46
 1
+ 110
+ 140
+ 158
+ 228
+ 1
+ 119
+ 144
+ 162
+ 20
153
131
 93
 72
56
11
09
+ 10
+ 11
+ 14
153
131
+ 13
+ 15'
+ 181 I +200
Degrees of Intensity.
38694
3
4
7
11
8
1
3
11
6
10
9
5
10
39680
2
4
10
7
2
2
10
4
9
9
3
43644
( 9)
(12)
(12)
7
1
1
10
5
10
5
7
3
(3)
47263
( 9)
(11)
(ID
(11)
( 7)
( 7)
10
7
9
9
2
55
55
3
1
Table XIV.—
Structure of
the Bands after
August 1901.
Mean
Adopted
dm
Degrees
of
T. M.
Intensity.
189
154
3
9
11
9
 133
 115
 94
 78
 65
J
2
 50
 14
10
5
10
9
3
8
5
+ 12
+ 38
+ 54
+ 80
+ 107
+ 119
+ 140
i
3
1
+ 158
+ 202
0*2 t.m. for a m , and 0*2 t.m. for A. TO . The observed degrees of intensity also agree
satisfactorily with the calculated ones, or their sums at those places where two or three
bands are superposed. The average difference, apart from signs, is 0'7 degrees of
intensity; 91 of the discrepancies lie between and 0"5, 60 between 0*5 and 1, 26
between 1, and 2 and 6 are greater. I consider it therefore proved that the bands are
in every way defined by \ n , the degree of intensity of their maximum, and the
quantities given in Table XIV.
13. Permanency of Structure. — Table XV. gives the number of observations of a
minimum, and the period during which it was observed. The bands A m = 3869 and
3968 were outside the range of the photoplates Nos. 8 to 17. Their position is never
theless well determined, since the plates Nos. 18 and 19 contain all the six minima, and
Nos. 20 and 21 each four. The first minimum appears to have been the most pro
nounced. In all, it was recognised 90 times, against 47 for the second minimum and
38 for the third minimum. Of the total of 177 minima, 158 belong to the 1901 plates.
The number of the minima that have been found seems to depend on the brightness of
the bands, and still more on the linear width of the minimum, which at wavelength
3870 was 0"23 mm., and at 5006 only 0'08 mm. In conformity with this, the
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 283
majority of the minima detected on the plates of 1902 belongs to the first two bands,
which at first were the brightest of the spectrum, and then second in brightness only to
the 5006band. They are present on each of the three photoplates on which they were
in focus, plates Nos. 21, 24, 26, 1902 January to November. During this time there
is also no change in the extent and the position of the maximum of the bands. The
evidence therefore points to the conclusion that from August 1 1901 to the end of 1902
the structure of the bands remained unchanged.
Table XV. — Table showing the Number of Photo Plates on which the Minima
have been measured, and the Periods to which they belong.
Km
Minimum 1.
Minimum 2.
Minimum 3.
of
Bands.
5
1
Period.
<3
Period.
CD
6
a
Sz<
4
Period.
38695
6
1901 Oct. to 1902
Mar.
4
1901 Oct. to 1902, Mar.
1901 Oct. to 1902
Nov.
39680
4
Oct. to
Jan.
6
Oct. to Nov.
4
Oct. to
Jan.
41034
7
Aug. to
Nov.
6
Aug. to 1903 Jan.
5
Aug. to
May
43410
8
Aug. to 1901
Nov.
7
Aug. to 1902 Nov.
5
Aug. to 1901
Nov.
43645
13
Ausj. to 1902
Jan.
9
Aug. to 1901 Nov.
7
Aug. to
Nov.
46126
1
Sept. 4.
3
Aug. to Nov.
1
Aug. 15.
46420
4
Aug. to 1901
Sept.
1
Aug. 26.
46878
9
Aug. to
Nov.
8
Aug. to
Nov.
47156
11
Aug. to
Nov.
47261
8
Aug. to
Nov.
5
Aug. to Nov.
3
Oct. to 1902
Mar.
48628
7
Aug. to
Oct.
2
Aug. 21, Sept. 4.
49594
4
Aug.
1
Nov. 13.
50072
8
Aug. to 1903
Jan.
3
Aug. to 1901 Sept.
1
Sept. 4.
A fourth minimum was measured twice in band A.,,, = 4364.5 on 1901, August 21 and 26.
14. Identification of the Bands. — The waveJengths A m of the zeros of the principal
bands are compiled in the first column of Table XVI. Five of these can with certainty
be identified as the hydrogen lines H^, H v , H s , and the two principal nebular lines. A m
is the wavelength of an arbitrary zero of the band, viz., approximately the mean of
the wavelengths of the three minima. I change the zero and make it coincide with
the wavelength \ of the line to which the band would be reduced under ordinary
conditions. Let [s] be the mean of the corrections for the orbital motion of the earth
on the days on which the photographs were taken, then
A , . r.i . A
X = Ao + %5()0 = X "' + W + rt '"4500
(13)
8a m = a n
4500,, r , . N
•a m =  T — (Am + LsJ A )
K = K
4500
8a m + [s]
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10).
43
284
PROFESSOR L. BECKER ON
Table XVI.
Sa„
Am
W
\
Hydrogen
and
Nebulae.
8om
38695
+ 01
39680
00
41034
+ 03
41018
+ 21
43410
+ 03
43407
+ 06
43645
+ 03
46126
+ 04
46420
+ 04
or (46353)
46878
+ 04
+ 04
(46338)
47156
+ 04
47261
+ 04
48628
+ 04
48615
+ 16
49594
+ 04
49590
+ 07
50072
404
50070
406
The five lines mentioned above give on an average Sa m = f 1*1 t.m. A calculated by
the third formula is comparable with the wavelengths of elements, and also with the
A derived from the March April plates (Table VII.). The motion of the new star in the
line of sight is here eliminated. If one should succeed in deriving from experiments,
or theoretical considerations, the wavelength A' of a certain point a Q of the band belong
ing to a line A' , a! can be calculated from a' = (A' — A / )4500/A / , and the ratio (a Q — a')/4500
is the ratio of the velocity in the line of sight and the velocity of light.
The residuals entered in the column headed " Difference " of Table XVII. exceed the
quantity that might be expected from the average error of A m , and perhaps the fact that
they rest on fewer minima than those of the brighter bands made them less accurate
than the latter. Some of the bands call for special remarks. The zero of the second
band lies 1*6 t.m. from the calcium line 3968*6, and 3*2 t.m. from the hydrogen
line 3970'2. Owing to the good definition of its minima and its isolated position, its
wavelength is one of the best determined of the spectrum, and its error is not likely to
be greater than the calculated average error. As r>a m cannot be so much in error, I
take the band to belong neither to calcium nor to hydrogen. The bright band whose
zero has the wavelength 4641*3 cannot be identified. This band overlaps the band,
A = 4611 '9, and only two of the three minima were measured. The zero would almost
agree with the bright hydrogen line 4633*8 if the observed minima were not the first and
second of the standard band, but the second and third ; an assumption which changes the
wavelength of the zero by the distance of two minima. This identification is bracketed
as an alternative, though it is questionable, because the first calculated maximum of the
band has not been observed (see Table XII.). The next band, A = 4687*1, is certainly
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 285
not due to the hydrogen radiation of wavelength 4684. Near A = 4725 , 3 is the
hydrogen line 4723  6, which in the MarchApril spectrum was identified with a band
Table XVII.—
Spect
rum
of Nova Perseifrom
1901
August 1 to 1903 January.
Nova Peksei.
Planetary
Nebulae.
Nova
Aurigse.
i
*0
Q3
O
S
Degree of Intensity
of Maximum of Band on
Photo Plate.
Io
Relative Intensities of the
Radiations.
(Campbell).
(Campbell).
o
U
60 a
a
hO
o
OS
I— 1
1901, October 31 to
November 13.
5
o
OS
o
u .
O x>
a
O £
OS
I— 1
o
<§
So
o
CO CO
^ 1— (
o§
OS
s
19
O
OS
O
t»
o> ^
o ,n
of
OS
A
I
\
I
38687
110
70
04
30
24
5
3868
4,5
39670
95
50
08
25
18
8
3969
4,5
396
05
41027
+ 09
31
13
07
08
7
5
4
8
41018
5,6
4098
2
43402
05
52
38
30
05
11
11
11
6
43407
5,6
4336
1
43637
101
77
23
20
20
9
43638
2,4
4358
8
46119
14
17
02
03
4
5
2
4
4610
0,1
460
1
46413
55
31
10
oi
11
9
5
2
or (46346)
4637
0,2
4630
7
46871
52
40
12
03
11
11
6
4
4687
2,5
4681
4
47148
65
26
05
oi
13
8
3
2
4715
2,4
471
1
47253
53
60
12
11
16
6
48620
+ 05
48
15
03
oi
10
5
2
2
48615
v.b
4857
10
49586
04
26
20
05
04
6
6
3
5
49590
v.b
4953
30
50064
06
49
35
25
10
10
10
10
10
50070
v.b
5002
100
Faint Bands.
38133835
05
6
3889  3952
02
02
1
3
3889
0,4
4027  4045
08
4
4026
0,4
4063  4080
06
oi
05
oi
2
1
3
2
41404165
05
02
2
2
42204265
05
05
05
03
2
2
3
4
424
4265
0,1
0,1
423
426
1
1
43004323
08
08
02
01
2
3
2
2
43824398
08
08
03
03
2
3
2
4
4390
0,4
438
1
4405  4488
13
15
07
04
3
5
4
5
44726
0,5
4466
1
4503  4590
09
10
07
03
3
4
4
4
4574
4597
4662
0,2
0,1
1,4
451
1
47474768
04
15
06
01
2
5
3
2
4744
2,4
47764786
10
05
10?
4
3
10?
47994810
06
05
3
6
4824  4840
03
06
1
3
48864900
oi
05
03
3
4
49174938
01
07
03
03
3
2
4
5031  5053
oi
05
03
2
2
of medium intensity. As neither the wavelength nor the intensity of the maximum
agree with those of the hydrogen line, it probably is not due to hydrogen, though the
286 PROFESSOR L. BECKER ON
possibility is not excluded that the MarchApril identification is wrong, and that both
the earlier and the later bands belong to the same radiations.
In the second half of the table appear the twelve corresponding lines which Campbell
photographed in the spectra of five planetary nebulae, and the range of their intensities
in these five spectra, I standing for " feint," and 6 for " very bright." Besides these,
there are only two lines, A. = 4662, intensity 1 to 4, and X = 4744, intensity 2 to 4,
which Campbell found present in each of the five nebulae. The first falls within the
range of the two bright bands X = 4642 and 4688 of the Nova spectrum, and if faint,
would be masked by them ; while the second is probably not represented by the faint
band 4747 to 4768 of the Nova spectrum lower down in the table. All the prominent
lines of the nebulae spectrum are present in that of the Nova, 3868 and 4364, in
addition to the principal nebular lines and the hydrogen lines, and their wavelengths
agree within their probable errors. I have already said that the second line of Nova
Persei could not be the hydrogen line 3970 '2. The planetary spectrum is not decisive
on this point. All the hydrogen lines are bright, and the intensity of 3969 fits into
their series, while its wavelength may be a t.m. in error. If it were the hydrogen
line, the Nova spectrum after January 1902 could be reconciled with it. It is possible
that a faint hydrogen band was superposed on the bright band X = 3967'0, and that
after January 1902, when the band faded and the measurements became difficult and
less accurate, its principal constituent was the band X = 3970. The only prominent
band of the Nova spectrum which has no counterpart in the nebular spectrum belongs
to wavelength 4725 '3.
15. Variation of the Bands and of the Radiations in Intensity. — The degree of
intensity ^ of the maximum of a band determines the intensity curve of the band. It
alone requires to be discussed. From 1901 August 1 to October 6 the observed values
of /x agree with each other within their probable errors, and 1 have combined them to
mean values. In this period the photoplates Nos. 8 to 17 were all taken at the same
angle of inclination. I have also combined the estimates made in three other periods,
using in each period photoplates taken at angles of 30° and 16°, and discarding those
estimates which belong to bands out of focus. The results, which are corrected for
the superposed bands, are tabulated in Table XVII.
By means of the formulae given in § 9 the relative changes in intensity of the radiations
which produced the bands can be calculated. Let a radiation of intensity i in the focal plane
of the spectrograph produce in time t a degree n of blackness on a photographic plate
whose sensitiveness is s for the wavelength of this radiation, and I designate these
conditions by (i, s, m, t), and let a radiation i produce in the same time t on the same
plate, for sensitiveness s a degree yw of blackness (%, s , /w , t), and for sensitiveness a' a
degree /*'„, (i , s, // , t).
I define sensitiveness by st = constant for the same intensity of radiation and the
same degree of blackness. i/i is the quantity wanted. I apply formula (5) to (i, s, n, t)
and (i , s, /*'„, t).
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 287
* Y = M_
/(/*'o) can be eliminated by /(/*<>)• I a Pply formula (2) to (i , s , n , t) and (i , s, m'o, 0»
which can be replaced by (i , s , n' , —), then
$0
(U) s /W
AA)
An)
Since the sensitiveness is constant for the same wavelength, I/I suffices for our
purpose. It appears in Table XVII. , where it is referred to the intensity of the radiation
at X =5007.
Compared with the MarchApril spectrum all the bright bands except the three H
bands, and perhaps the bands X = 46346 and 4611*9, have come into existence, or have
grown much in intensity. The radiation at X = 43637 is twice as intense as that of
the three Hbands, which in the spring were the most prominent bands of the spectrum.
The other bands observed in the spring have almost entirely disappeared. None of the
radiations faded at a slower rate than the nebular radiation X = 5007. Several radiations
began to decrease already in October, and others started in November. The decline of
the bands X = 43637 and 4725 in the two months 1901 November 13 to 1902 January
12 is further remarkable. Four radiations faded at the same rate, H s , 4612, 4959, 5007,
and perhaps H y , while H 3 certainly decreased at a much greater rate in October. I may
mention that I calculated s /s by (14) from estimates of degrees of blackness made on a
photograph of the sky spectrum. The values i/i calculated by (15) show that, with the
only exception of the radiation at X =3869, that at X = 5007 was the most intense
already in August 1901.
In October and November 1902 the relative intensities agree better with those of the
nebular spectrum than before. It must of course be borne in mind that the spectrum
on the last plates was extremely difficult to see, the intensity of the maximum of the
band X = 5007 being only of degree 1, and that the figures belonging to that period
are only a rough approximation. The trend of the table is certainly to show that the
intensities are approaching those of the average nebular state.
16. The Faint Bands. — One may conclude by analogy that the faint bands would
be reduced to lines under ordinary conditions. In that case, on account of their
breadth, several must be due to multiple lines. On a whole they agree fairly
well with the maxima of the intensity curve observed in March and April. Considering
the uncertainty of the wavelengths of these faint bands, about 5 t.m., a convincing
proof as to their origin cannot be brought, though it is probable that they are the
remnants of the bright spectrum in the first months. It may be mentioned that the
hydrogen lines given under B in Table VII. also explain them, provided seven of them
be excluded.
288
PROFESSOR L. BECKER ON
17. Last Visual Observation of the Spectrum. — On March 3 1903 I inspected the
spectrum of Nova Persei in the focal plane of the spectrograph without using an eye
piece, a method which I usually employed prior to the exposure, to make sure that the
proper star had been set on the slit. I saw only one bright spot in the whole range of
the spectrum which coincided with the air band at 5004. Several times I gained the
impression that there was a faint spot near the place of the magnesium line 4481. The
comparison was made in this way, that when the eye had been fixed on the spot the
spark was switched on for an instant.
18. Curious relation between A 1 of Four Prominent Lines. — The wavelengths of the
zeros of the brightest bands are 3869, 3967, 4364 and 5007. The oscillation frequencies
of the first, third, and fourth zeros almost form an arithmetic series, which, continued
to the less refrangible side, gives the wavelength of the helium line D 3 , a line which
was measured by others in the spectrum of the new star, and also belongs to the nebular
spectrum. In the following table I give the wavelength of the helium line, Keeler's
determination of the nebular line, and my determination of the other two lines, reduced
to the two nebular lines as standards.
The formula
A" 1 = 170142 + 2957*6»  5'5?t 2 , n = , 1 , 2 , 3
determines A as entered in the last column. The agreement is perfect. Should this
be merely a casual coincidence ?
A
 Vacuum.
A
Difference.
Calculated A.
38692
43643
500705
587587
258378
229068
199663
170142
29310
29405
29521
38692
43643
500705
587587
19. Similarity of the Structure of the Bands in MarchApril 1901 and after August
1901. — I add Sa m = + 1.1 t.m. to a m of Table XIV., which reduces them to the same zero
as was employed for the March April bands. Both bands are given in Table XVIII. , and
also on Plate II. I include the second minimum of the MarchApril band, though it
rests on only two single measurements in two bands, because it seems to fill up a gap in
the order of the minima. The extent of the maximum and the position of the minima
agree with each other. There is only this difference, that while the March April band
declines to nothing from a =  12 to  73 t.m., and from + 13 to + 56 t.m., the later
bands fade abruptly on 6 t.m. Between April and August the ends of the bands have
therefore decreased at a greater rate than the central maximum portion. I repeat again
that from August 1901 to January 1902 no change took place in the structure, and
that the extent of the maximum remained unaltered during 1902. It appears that the
spectrum converges towards a nebular spectrum, in which each line is broadened 27 t.m.
THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 289
Table XVIII. — The Structure of the Bands in MarchApril 1901, and after
August 1901, referred to \ as zero, and reduced to X = 4500.
Structure of the Bands.
Reversals.
MarchApril.
After August.
MarchApril.
After August.
.£ m °
^ .
a
Intensity.
«o
Intensity.
«o
Q * <&
«o
o "3 "3
T.M.
T.M.
T.M.
*" G %
Jo?
"a ° «
P MM
fc3 £
Number
of Plates.
T.M.
8 § v
5 c
Index
Number
of Plates.
CO —
pq.S
73
55
5
43
5
6
315
7
239
" 0*
 22  8 *
8
3
218
8
175
9
178
 166
)
£k
3
} 143
155±
2
1
144
11
J
3
12
13
 122
104
 83
9
11
9
1
2
 58
 67
7
 54
 48
7
5
 53
89
8 to 24
 38
13
 39
10
+ 12
 03
7
5
+ 22
5
4
+ 10
48
9 to 27
+ 32
125
+ 23
+ 49
10
9
+ 68
+ 65
7
3
+ 81
6
4
+ 78
38
9 to 26
■f 95
125
+ 91
8
+ 124
+ 118
7
5
+ 127
3
3
+ 124
2
lOandll
+ 130
+ 130
7
115
+ 15T
3
+ 168
10
+ 169
1
+ 201
8
+ 213
+ 251
7
55
+ 317
35
i
+ 34
2 5
+ 364
1
+ 405
J
05
+ 56
Sharp Fraunhofer line, 1901, March 1820 and March 25.
"290 THE SPECTRUM OF NOVA PERSEL AND THE STRUCTURE OF ITS BANDS.
20. Results. — 1. The spectrum consists of a line spectrum in which each line is
broadened into a band, the broadening being proportional to the wavelength of the
line and independent of the element. Tables V. and XIV. give the common structure
of the bands.
The position of the maxima and of the minima or reversals remains unchanged
during the whole period 1901 March to 1902 November. (See Table XVIII. and
Plate II.)
2. The intensity curve of the spectrum in March and April is satisfied by the
hydrogen and helium lines, some of which vary in intensity during this period. (See
Table VII. B, and Plates I. and II.) It is probable that the spectrum is due to
hydrogen and helium.
3. From August 1 1901 to the end of 1902 the bands belong to the lines of the
spectrum of planetary nebulae, and their relative intensities converge towards those of
the average nebular spectrum. (See Table XVII. and Plate II.) Probably the March
April spectrum is also faintly present during the whole period.
I wish to acknowledge the help I have received from my Assistant, Mr James
Gonnell, who attended to the guiding of the telescope and plotted the curves given
in the plates accompanying this paper.
Trans. Roy. Soc. Edin
BECKER: ON THE SPEC^u
Intensity curves of the spectra of Nov
calculated
servi
4410
10YA PE RS EI. Plate I.
Vol. XLI.
rfarch 3, March 1820, and March 25
5016 —
1 i ! i i  i i i i  i — i — i — i — f — i — i — i — i —  — i — i — i — i — I i i i t i i — i — i — i — 1 — i — i — i i i r
4600 4700 4800 4900
5000
A RITCHI1 ■■  "■■ El '.'
.
I
Roy. Soc. Edin. Vol. XLI.
BECKER: ON THE SPECTRUM OF NOVA P E R S E I. — plate ii.
Ptattl. 1901 March 3.
CD
P
co
o
3
3"
b.
o
o
ST e
i
(0
Intensities
0—N(*J^Ulm>Jooioo"'^ u
.f^OO^tJ^O^^JOOtOi
o _
o _
z
p
*
p
3
to 1
M> —
o
o ~
ft* z
o
o
o _
—

*k 
o 
—
—
_
*. I
o
3
o _
o
—
—
o —
° — 1
& 
o ~


° z

z
o

"Z
00
o
o ~
£>. Z
<x>
o
o 
_z
S_z
3 —
o 
_
ISJ —
o Z
o — Z
o _
tr» Z
° z
o
o —
z
o
o ~
~.
!
o
CO
■o
a>
o
3
o
o
•<
P
a>
a.
t=
o
CD
Q.
CO
3
CO
lt
«<
O
c
s
<.
CO
CO
o
o
CO
a
o
s.
*
.
Roy. Soc. Edin, Vol. XLI.
BECKER: ON THE SPECTRUM OF NOVA P E R S E I. — plate hi.
ngtks A m of zero of bands
38695
Curves of observed intensity of 12 bands in the spectrum of Nova Persei.
1901 August to November . abscissae a™ (see a 11)
M
3968
41034
43410
43645
46126
46420
4687i
47261
48628
49594
50072
TenDimetr es
I ' ' ' ' I ' ' ' ' I I  I II I II  ' I I I i I I   I I i I i i i i i i i  i i i i i i
20 15 10 5 ' ' O +5 +10 +15 + l 2Q
( 291 )
XI. — The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part I.
Structure of the Resting and Dividing Corpuscles. By Thomas H. Bryce,
M.A., M.D. (With Five Plates.)
(Read January 18, 1904 ; MS. received March 19, 1904. Issued separately November 19, 1904.)
The material for the observations recorded in this paper has been kindly lent to
me by Mr Graham Kerr, Professor of Zoology in the University of Glasgow. It
consisted of some of his beautiful series of cut embryos, and of some freshlysectioned
material which I stained specially for the purposes of the research.
The blood corpuscles of the embryo Lepidosiren are exceptionally favourable objects
for the study not only of the morphology of the blood, but also of cell structure. The
karyokinesis in the red corpuscles presents features of considerable interest — and the
phenomena are presented to the observer on such a scale as to render them almost
diagrammatic.
In the present paper I shall deal with the structure of the corpuscles and the
mitotic phases in the erythrocytes, reserving for a future communication the results of
studies on the origin and histogenesis of the elements.
Methods.
For the study of the dividing red corpuscles I selected a stage in which the embryo
was small enough to have permitted perfect penetration of the fixative fluids, and yet
sufficiently advanced to have its cells free of yolk.
The stage selected was that represented in pi. x. fig. 32 of Mr Graham Kerr's
memoir* on "The External Features in the Development of Lepidosiren paradoxa
(Fitz.)," a larva twentyfour days after hatching.
The embryos chosen had been fixed in sublimoacetic fluid, and the fixation leaves
nothing to be desired.
The sections were cut at 10 m, which was rather thick for some points, but the
nature of the material, owing to the mass of yolk, did not permit of thin sections.
The stain employed was in the first instance iron hsematoxylin, with a counter
stain of eosin. It was, however, discovered that even at this early stage several
varieties of leucocytes were present in the blood, and for the study of these a stain of
methylene blue and eosin, and the mixture of Ehrlich known as Triacid were employed.
The best results in some respects were obtained with the first named, especially for
the centrosome of the erythrocytes, but for the resting red corpuscles and the leucocytes
the methylene blue and eosin gave a finer differential colorisation.
* Phil. Trans., vol. cxcii. B. 182, 1899.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 1 1). 44
292 DR THOMAS H. BRYCE ON
The two dyes were applied successively, and not in a mixture, according to a method
described by Dr Gulland, and communicated to me by Dr Goodall. The sections
were stained first for about five minutes with a saturated watery solution of eosin, and
then after washing, with a saturated watery solution of methylene blue for two or
three minutes. They were then washed and differentiated if necessary in 90° alcohol,
dehydrated and cleared in pure xylol.
The sections stained by this method are as bright after nine months as they were
at first.
I. Structure of the Erythrocytes.
(a) Resting Corpuscles.
The red blood corpuscles are oval biconvex discs, varying in size from 42 to 50 m in
length, 30 to 36 m in breadth, and 12 to 15 m in thickness. The nucleus occupies the
centre of the disc (PI. I. fig. 1, PL IV. fig. 32). It is also oval in shape, measuring
20 to 27 m in length, 12 to 15 m in breadth, and 9 to 12 m in thickness.
(1) Cytoplasm.
The corpuscle is surrounded by a delicate membrane. The cell body shows a
peripheral ring or band, within which there is a coarse meshwork structure. The
meshwork is not very regular, but the thickness of the sections intensifies the
appearance of irregularity. The meshes are from 3 to 4 m in diameter. The whole
reticulum centres on the nucleus, having a general radial direction from nucleus to
periphery. At the nodal points there are strongly refractile granules of considerable
size.
In some corpuscles the fibrillse of the reticulum in the central nuclear portion of the
corpuscle are arranged as parallel running threads between the nucleus and the periphery,
but it is not quite clear how far this is a normal appearance.
The staining reactions of the meshwork are as follows : — With iron hsematoxylin
it is grey, while the microsomes are black (PI. I. figs. 1 and 2) ; with methylene blue
and eosin, the meshwork stains bright red and the microsomes are dark red spots ;
with triacid it is yellow, and the microsomes stand out as darker yellowishbrown points.
In some favourable stainings with the lastnamed mixture the alveoli had a faint pink
tinge.
In all the larger corpuscles there is a large vacuole, with structureless contents,
showing no differential reaction to any of the stains used.
Pound the equator of the cell there is a remarkable band about 3 m in diameter. It
forms a complete peripheral ring, when the corpuscle is seen on the flat (PI. I. fig. 1,
PI. IV. fig. 32). In the fixed cell its appearance is distinctly fibrillar. The fibrillse run
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 293
concentrically ; and though they seem for the most part parallel, there is a considerable
amount of apparent crossing and recrossing.
In profile views (PL I. fig. 2 and PI. IV. fig. 33) there is to he observed at each
end of the corpuscle an area free of reticular formation, but occupied by a number of
fine points arranged generally in a straight line. These I take to be the cross sections
of what appear to be fibrillae seen on the flat.
Such a peripheral ring has been described in the red blood corpuscles of the chick
embryo by Dehler,* and in amphibian corpuscles by Nicolas + and Meves. The latter
has demonstrated that in Salamandra the ring is fibrillar, consisting of very fine threads
running parallel to one another, or a single unbroken thread developed into a skein in
the wall of the corpuscle. This is displaced inwards at the beginning of mitosis, under
goes a loosening, and then disappears as such, its substance being apparently employed
for the formation of the achromatic figure.
The structure thus described by Meves is evidently of exactly the same nature as
the band in the Lepidosiren corpuscles, but he finds no network such as I have described,
and the question here arises whether that structure is not a precipitation product.
A reticular or meshwork structure has been described in amphibian erythrocytes by a
number of authors (Leydig, Frommann, Auerbach, Foa, and others), but it has been
variously interpreted. GiglioTos § figures a reticulum identically like that I have
described, in the erythrocytes of the lamprey. What I have named the microsomes he
calls hsemoglobigenic granules. Recently Ruzicka  has represented the corpuscles of
Rana as having a reticular structure closely resembling that seen in the Lepidosiren
cells. Butschli,1T on the other hand, attributes to the outer portion of the corpuscles
in Rana an alveolar structure bounded by a distinct membrane. Within this outer
zone is an inner girdlelike zone of finely meshed internal protoplasm, while the central
nuclear portion is occupied by a space containing stuctureless enchylema, in which
there are radiating tracts of protoplasm.
I do not propose to discuss the history of opinion on the structure of the red discs,
but I may mention that Rollett ** in a recent paper concludes for an alveolar stroma,
while WEiDENREiCH'stt recent observations support Schafer's conclusions (published
in Quain's Anatomy), that the contents are fluid and structureless, enclosed by a
membrane. In this case I feel no doubt of the existence of a membrane, but reserve
is necessary as to the reticulum. It must be noted, however, that I am dealing with
young corpuscles. JJ
* Archivf. mikr. Anat., Bd. 46, 1895. t Bibliographie anatomique, 1896.
I Anat. Anzeiger, Bd. 23, 1903. § GiglioTos, Mem. Accad. delle Sc. Torino, T. xlvi., 1896.
 Anat. Anzeiger, July 1903, Bd. 23. 1 Protoplasm, etc., English trans., 1894, p. 125.
** Pfluger's Archivf. Physiologie, Bd. 82, 1900. ft Arch.f. mikr. Anat., Bd. 61, 1902, p. 459.
H Meves, in a paper published since this paper was written (Anat. Anzeiger, vol. xxiv. No. 18), holds that there
is no membrane in the amphibian corpuscles. The peripheral ring of fibrillse is the only structural arrangement in
Salamandra, but he states that in Rana there is, in addition, a ' Fadenwerk,' which is collected further round the
nucleus, especially at its poles, and he quotes Hensen (Zeitschr.f. uriss. Zool, Bd. U, 1862) as having described in the
corpuscles of the Frog a granular material round the nucleus, from which threads pass to the periphery.
294 DR. THOMAS H. BRYCE ON
The Lepidosiren corpuscles thus resemble those of Salamandra in the possession of a
very distinct equatorial band, but in their reticular structure they seem to correspond
more to the description given of the corpuscles of the Frog. Taking all the possibilities
into account, I adopt the view that the reticulum is not an artifact, but that it
represents a protoplasmic framework. This is possibly alveolar in arrangement, but
it is clear that the meshes of the reticulum exceed considerably the limit laid down by
Butschli for the true protoplasmic alveoli, and greatly exceed those of the optical
reticulum seen in the protoplasm of the leucocytes. The erythrocyte is a much
differentiated cell, and the structure described is evidently a secondary one. The
whole protoplasm is fibrillar, but the framework is not necessarily fibrous or fixed. I
believe rather that it is colloidal. I derive it from a vacuolated condition, in which
the active protoplasm (Hyaloplasm) is greatly reduced, and it may well be that an
original alveolar arrangement has been lost by the breaking through of mesh walls.
The peripheral band must be either the cause or the consequence of the shape of the
corpuscle. It disappears when the disc begins to round up for division. This suggests
the possibility that the appearance is due to a massing of the mesh walls. Further, in
the angular interval between the upper and under layers of the membrane round the
equator, there is a space (fig. 2, PI. I.) occupied by the fibrillae of the ring cut across.
When the corpuscle rounds up, this space disappears, and the band is replaced by a
reticular formation.
These considerations, combined with observations on young corpuscles, incline me
to the view that the ring may rather be the consequence than the mechanical cause
of the shape of the corpuscle, but the matter will come up for discussion again in the
second part of these studies, when I am in the position to deal with the histogenesis of
the cells. #
The question here arises whether the corpuscles which have assumed the biconvex
disc shape are capable of division. Besides the corpuscles with oval nuclei, there are
others with round nuclei, and a smaller cell body showing a finer reticular structure.
These do not assume the disc shape, though they are oval in form. They are found in
active division. In the second part of this memoir I shall discuss the relationship
between these two forms. Meantime it has to be determined whether both classes of
cells are dividing elements. In the later stages of mitosis there is little to distinguish
the one class from the other, for all dividing corpuscles are spherical. Variation in
the size of the chromosomes would indicate a derivation from a coarser or finer
chromatin network, and the roundnucleated corpuscles have distinctly a finer network
of chromatin than those with oval nuclei. Direct observation, however, shows that by
far the greater number of nuclei showing prophase stages are oval in shape, and between
* Meves, in a recent paper cited in the note to page 293, concludes that the band is the cause of the biconvex
shape of the corpuscle. His explanation of the mechanism does not seem to apply very satisfactorily to the Lepidosiren
corpuscles, but I must postpone a discussion of the question until all the stages in their histogenesis have been worked
out. It seems to me that it is only by a study of the developmental stages that the significance of the band or ring can
be determined.
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 295
the rounded daughter corpuscles, and the biconvex disc shaped corpuscles, are to be seen
all varieties of intermediate stages. These must necessarily be corpuscles either
assuming the disc form or rounding up again for division. These intermediate forms
sometimes possess two centrosomes ; sometimes no centrosome can be demonstrated.
The corpuscles showing early prophase stages of the nucleus always have two very
distinct centrosomes, and they are either quite spherical (fig. 8, PL I.) or are oval and
often irregular, showing, according to the plane of the section, one or two lateral
projections (PI. I. figs. 3 and 4).
Corpuscles such as that drawn in fig. 4, PI. I. are frequently met with, in which two
very distinct centrosomes are present, although the oval nucleus shows still a coarse
network.
I believe I am justified in stating that, while it is possible that the corpuscles with
vesicular nuclei may not divide, at any rate the smaller discshaped ones represent the
resting phase of the dividing cells.
In the resting stage, if this be so, no centrosome is present in any form in which
it can be distinguished from the microsomes.
(2) Nucleus.
As mentioned above, the nucleus is an oval body. It has a very coarse chromatin
network (PI. 1. figs. 1 to 5), with large karyosomes close packed. In a considerable
number of corpuscles the nucleus is to all appearance a solid mass of chromatin.
The reactions of the nucleus to the various dyes is interesting. In iron hsema
toxylin material the chromatin holds the stain with great persistency, so that the
erythrocyte nuclei are still intensely black after all the other nuclei have completely
surrendered it. With methylene blue and eosin, the network has a blackish violet
colour, quite different from the lighter violet of the nuclei of the leucocytes, and again
from the pure blue of the nuclei of the cells of the mesenchyme.
The chromatin network again selects the orange from Ehrlich's mixture, and has a
golden colour. The alveoli are occupied by a delicate green staining, but no linin
threads can be made out. In a successfully stained specimen the chromatin of the
mesenchyme nuclei selects the basic dye, and their green colour contrasts with the
golden yellow of the nuclei of the red corpuscles.
Notwithstanding this behaviour to the dyes, the rounded masses in the nuclei are
not true nucleoli, but karyosomes,* or at any rate they are local accumulations of the
same substance as forms the intervening bars, and, as later, is uniformly distributed
along the spireme thread.
(b) Mitosis.
As I have already stated, no centrosome is to be seen in any recognisable form in
any of the resting corpuscles, large or small.
* Of. Pappenheim, Virchow's Archiv, vol. 145.
296 DR THOMAS H. BRYCE ON
The first evidence of the onset of mitosis is the formation of a bulging of the central
nuclear portion of the corpuscle on one side. In this projection are seen in the vast
majority of cases two centrosomes lying side by side, and close to the surface of the
corpuscle, and remote from the nucleus (PI. I. fig. 4). Each centrosome is the focal point
of farreaching radiations, which are clearly directly continuous with the reticulum of the
corpuscle. They have every appearance of being simply a radially disposed portion of
the general network. The centrosomes are not connected directly by intervening fibres.
In fig. 4, PI. I. an appearance seen in that, as well as other corpuscles, is suggested. On
the left of the nucleus the meshes of the network appear drawn out towards the site of
the centrosomes, and the radial fibrillae can be traced far out forming the walls of the
meshes of the network. This appearance is transitory. In the next stage (PI. I. fig. 8,
PI. IV. fig. 34) the lateral wings have been drawn in, and the corpuscle has become
spherical. The radiations are confined to one pole of the cell, the centrosomes remain
ing near together and close to the surface.
(l) Structure of Centrosome.
The structure of the centrosome varies according to the character of fixation
and the manner of staining. In iron heematoxylin sections the appearances depend
on the degree of abstraction of the stain. When much of the stain is left, the body
is a very large one, and the black colour is even continued out along the radial
fibrillse. When the decoloration is carried far, there is a much smaller dark point
in the centre of a halo staining red in preparations counterstained with eosin. The
fibrillse spring from the circumference of this halo (PI. II. fig. 13). This is clearly
an instance of concentric decoloration, and the black spot is not a true centriole in
Boveri's # sense. I have not been able to demonstrate a single such centriole or
pair of centrioles at any stage of mitosis, but frequently the centrosome has the
appearance of a grey spot, containing a group of centrioles. A slightly lobed
appearance of a solid centrosome points to the same structure, even though no
separate granules are to be made out. The question arises whether this is a
' fragmentation ' of the centrosome (Boveri) * or the true structure.
With the other dyes used the centrosome is not so vividly differentiated as with
iron hsematoxylin, but in view of the tendency of that stain to mask a finer structure
by remaining lodged between the smaller elements, a truer picture is perhaps obtained
by their use. In methylene blue and eosin sections the centrosome is a red area
occupied by fine granules of the same size as the microsomes, but darker in colour,
having a neutral tint — an appearance very possibly due to their being massed together.
In the same way in triacid preparations the centrosome is yellow, with brownish yellow
granules.
* ZellenStudien, Heft iv., " Uber die Natur der Centrosomen," 1900.
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 297
(2) Origin of Centrosome.
The origin of the two centrosomes is very difficult to arrive at. I have seen
only a very few corpuscles with a localised radial disposition of the reticulum in
which there were not two centrosomes, either together in the same, or one in each
of the adjoining sections of the series, and these few cases are difficult of interpretation.
There is not a single clearly defined centrosome, but an area (fig. 3, PL I.) on
which the radii converge. This is occupied by granules in every way similar to the
microsomes. The extreme rarity of this stage, if it be a stage, shows that it must be
a shortlived one, and that almost immediately the centre of activity is duplicated.
It is clear that the centrosome or centrosomes described have no direct relationship
to that of the previous division. The resting cells have no centrosome, and no
granules distinguishable from the microsomes which can be recognised as centrioles.
Further, the new centrosomes do not appear where the old disappear, and therefore,
unless on the purely theoretical assumption that the centrioles are scattered in
the protoplasm, and though indistinguishable retain their identity,* to become the
new focus or foci, the centrosome must be considered to appear de novo. That
this is actually the case is strongly supported by instances such as these figured
on PL 1. figs. 5 and 6, in which two centrosomes are seen so far removed from
one another that it is hardly possible to believe that they have not appeared quite
independently of one another. It is remarkable, however, that in later stages, when
the centrosomes are very far apart, presumably successors of a stage such as figured
(PL I. fig. 6), they are still single. I have seen no multipolar figures, and in such
cases there is a suggestion that the chromosomes are forming themselves into two
groups round the two asters. I have seen only a small number of such figures, but
even the one or two I have seen seem to prove that the two centrosomes may appear
independently ; and the fact that the independent centrosomes do not divide and
form multipolar figures further suggests the possibility, in the absence of any stage
in which a single definite centrosome can be confidently asserted, that the two
adjacent centrosomes are independent from the first — that is, as definite stainable and
recognisable foci.
(3) History of the Nucleus during Mitosis.
It will not be necessary to deal in detail with the history of the chromatin as
it presents only the wellknown evolutions ; a few points only require to be
mentioned.
The spireme thread is not beaded ; that is, there is no distinction between a linin
basis and chromatin granules imbedded in it. The whole thread stains uniformly.
In this respect it differs from the thread seen in the prophases in the nuclei of the
* Meves, Verhand. anat. Gesellschaft, 1902.
298 DR THOMAS H. BRYCE ON
leucocytes. The longitudinal splitting takes place early. The Vs are unequal, with
one short and one long leg. The latter in the metaphase is of such length that
when all seen in one section it extends round a third of the circumference of the
cell. This makes the metaphase figures so complicated that I cannot be certain of
the number of chromosomes.
In the late anaphases the chromosomes are merged again into a seemingly solid
mass of chromatin, which no amount of extraction will resolve into separate elements.
The long tails are gradually drawn into the common mass and an oval solid nucleus
is formed. In many resting cells, as mentioned above, the nucleus has the same
character, and the appearances point to the coarse reticular stage being reached by
a sort of vacuolation. Throughout all the phases the chromatin retains the staining
reactions described for the resting nucleus.
(4) History of the Achromatic Figure.
At the stage at which we left the centrosomes when they lay close together,
and the corpuscle has rounded up for division, we noticed that there were no direct
connecting threads between them. On their outer sides the radiations are strong
and join the general reticulum. As the centrosomes draw apart (Plate I. fig. 9) it
becomes clear that there are still no fibres directly joining the centrosomes, and
that the radiations are stronger on the side of the nucleus. Both at the equator of the
spindle figure and where the radiations of the asters meet, the fibres seem to branch and
anastomose. I think the appearances are in favour of an anastomosis rather than of a
mere crossing of the fibres ; one never sees a loose end at any stage of the process.
When both centrosomes are sharply in focus at the same time, the axis of the
spindle system is seen to be occupied only by a faint system of branching and
anastomosing fibrillae.
There is, strictly speaking, no ' central spindle ' spun out between the centrosomes,
but only two systems, mainly of mantle fibres, which join one another round the equator
(PL I. figs. 11,12). In a cross section of the metaphase figure there is no core of fibres
representing a crosscut central spindle in the heart of the equatorial crown ; only a few
fine fibrillae are to be made out.
The appearances point, not to any new formation of radiating fibres, but to a con
version, step by step, of the general network into radiating tracts, until it has all been
drawn into the opposing systems, and the achromatic figure comes to be placed sym
metrically in the corpuscle.
As the daughter chromosomes move apart, the axis of the spindle system is seen to
be occupied by looselyarranged, irregularlydisposed fibres ; and as the anaphase pro
gresses, the ' subequatorial fibres ' (Meves) come out more and more clearly, while
the axial system becomes more loosely arranged (PI. II. figs. 13, 14, 15 ; PI. V. fig. 39),
until we have a central space traversed by coarse muchbranched fibres, and bounded
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 299
laterally by very distinct fibrillse. These branch, and the branches join at the equator
those of the fibres of the opposite astral system, while the outermost threads abut
against the cell membrane, and those from opposite poles are seen to meet at the point
where it is becoming infolded (PI. II. fig. 15 ; PI. V. fig. 39). There is no thickening of
the membrane at the point of infolding.
I have shown that at all stages the axial system of fibres is very feebly developed.
The contortion of these fibres in the anaphase cannot be due to any ' pushing ' force
exerted along them, but rather I believe to the accumulation at, or determination
of fluid to the equator of the corpuscle. In some preparations I have seen an actual
vacuole occupying the spindle axis, as if the protoplasmic threads had been wholly
withdrawn towards the poles.
The subequatorial fibres become more strongly marked at this stage, and it is
certainly suggested that the lines of force are now directed on the cell periphery, and
the picture gives the idea that the force that is exerted by or along the lines of these
threads is rather a tractive than a pushing one. The determination of fluid to the
equator seems coincident with the passage of the chromosomes to the spindle poles.
It is to be noticed that the distance between the spindle poles is distinctly increased at
this stage.
Stages intermediate between that represented in PI. II. fig. 15 and that shown in
fig. 17 are rare, suggesting that once the infolding is produced, the cell division is
quickly completed.
The subequatorial threads, still attached to the cell membrane at the bottom of the
furrow, come to be stretched in a straight line between the spindle poles (PL V. fig. 40),
and at a later stage (PI. II. fig. 16 ; PL V. fig. 41) form, with the loose fibres in the axis
of the spindle, an hourglassshaped system of fibrillse. These are grouped apparently in
bundles, which contract into the ' midbody ' when division is complete. This has not
the ring form seen in some cells, but is a large single body, probably formed from the
smaller single granules on the bundles of threads of the previous stage (PL II. fig. 17).
It becomes drawn out into a longish thread when the daughter corpuscles separate from
one another (PL II. fig. 18 ; PL V. fig. 43).
The centrosome undergoes little increase in size during mitosis. There are no
phenomena comparable to the enlargement of the sphere which occurs in dividing ova.
In the late anaphases it is drawn out somewhat tangentially, and in the telophases it
begins to dwindle. It lies in the hollow of the reconstructing nucleus and is difficult to
detect, but in oblique sections it is seen standing out clear of the nucleus ; and in such
sections, although I have given much attention to the point, I have not been able to
convince myself that it was in any case duplicated (cf. fig. 17, PL II.).
In fig. 18, PL II. an appearance suggestive of a division is drawn, but careful
examination proved that the radiations were all focussed on one point, and that the
appearance was an accidental one, due probably to defective fixation. Similar deforma
tions of the reticulum are met with in other cells removed from the centrosomal area.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 11). 45
300 DR THOMAS H. BRYCE ON
I am obliged, therefore, to conclude that, as a recognisable structure, the centrosome
disappears completely when mitosis is over, and that, in the absence of any proof that
a contained centriole divides in the telophase to survive to the next generation, the
centrioles also disappear as such, whether they be twin centrioles or a group of centrioles
that could be supposed to persist.
Since the demonstration by Wilson,* following the earlier observations of Morgan,
that centrosomes arise de novo in the echinoderm egg during artificial parthenogenesis, a
conclusion such as I have come to will seem less improbable than it would have some
years ago.
(c) Interpretation.
As stated above, the conclusion was arrived at that the meshwork seen in the fixed
corpuscles represented a protoplasmic framework in the living cells. Certain features
of the resting cells, and certain appearances observed during mitosis, suggest that the
protoplasm is of a specially viscous or ductile nature. The early history of the
achromatic figure and of the centrosome preclude the application to this particular case
in sensu stricto of either the fibrillar hypothesis (Van Beneden), or of the doctrine of
the organic radii (Heidenhain). Both involve a structure of the resting cell which
does not exist in the erythrocytes. In the conceptions of Khumbler,! however, I find
room for a free formation of the centrosome ; and the interesting feature of this case is,
that the theoretical conditions of his model of cell division are fulfilled more closely
perhaps than in any hitherto described.
The general reticulum is in the resting cell centred on the nucleus. It is under
some degree of elastic tension, but the focus of that tension is not a centrosome, and
therefore the conditions are not such as represented in Heidenhain's \ model. On the
appearance of the centrosomes, the reticulum begins to show a new disposition. It is
now centred on these bodies, and round them is converted into radially directed
threads. This radial arrangement of the reticulum is probably brought about by
the withdrawal of the mesh walls circumferentially disposed into those radially dis
posed to the centrosomes.
Apart altogether from the why and wherefore, the centrosomes and their radiations
are a manifestation of a tendency of the protoplasm to retract or concentrate itself at
two focal points. The first effect of the retraction is the rounding up of the corpuscle ;
the second effect is the separation of the centrosomes.
When there are two centrosomes at some distance apart (PI. I. figs. 5 and 6), the
reticulum becomes converted into a symmetrical aster round each. When they lie close
together, the asters are not symmetrical, for between them the protoplasmic material is
limited, and is in large measure retracted on to the opposing centres. The progressive
condensation or retraction of the threads on the outer sides being thus in excess of that
* Arch.f. Entwickelungsmelc, Bd. xii., 1901. t Ibid., Bd. iii., iv., xvii.
X Ver. anat. GeselL, Berlin, 1896. Arch.f. Entvrickelunysmek., Bd. i.
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 301
between the centres, the result necessarily is to cause the centres to separate from one
another.
Thus the separation of the centrosomes cannot here be clue to the growth of a central
spindle between them, but more probably to the contraction or retraction of the astral
rays on their outer sides.
When the centrosomes have drawn apart to such a distance that the forces of which
each is the expression can dominate half the cell, the whole protoplasm arranges itself
symmetrically round them, and a position of equipoise is reached. The cytoplasm is
already divided into two exactly equal portions, with a neutral zone between.
When the nuclear membrane disappears on the side next the spindle, it is noticed
that the spindle system is disproportioned. The greater development on the nuclear
side is possibly due to the taking up by the two centres of nuclear substance, rather
than to an increase in the growth of the astral rays on that side, as has been suggested.
It has been a point much discussed, how far the nuclear substance shares in the for
mation of the spindle system. In this case, while the system is formed apparently
wholly in the cytoplasm, it seems almost certain that the achromatic substance of the
nucleus is also drawn into and divided in it.
The arrangement of the spindle fibres in the anaphases is much like that of the outer
polar fibres in Boveri's figures of the dividing eggs of Ascaris* but there is no plate at
the equator. That the chromosomes are separated by a pushing force on the central
spindle is excluded here by the absence of a developed central mass of fibrils.
The subequatorial fibres become exposed on the separation of the daughter chromo
somes, and I believe they are related here to the division of the cell body, not in virtue
of a pushing or expansive force, as Meves described,t but of a contractive force. In
fig. 15, PI. II. the condition of things is pretty clear. In the axis of the spindle system
there is a very loosely arranged mass of fibres, with large spaces between the threads,
while peripherally, from under the reconstructing nuclei the subequatorial fibres extend
towards the equator, and are there continuous with the cell membrane. The threads
from the opposite poles meet exactly at the equator on the surface of the cell. At this
stage the spindle poles are separated from one another, whether by a determination of
the enchylema, or substance from the contracting daughter nuclei, to the equator or
otherwise, and the consequence must be to put the longest subequatorial threads, i.e.
those reaching the surface at the equator, on the stretch, and if they be of a colloid
nature, they will, by their elastic tension, tend to retract on to their centres.
Thus we have produced a disposition of the protoplasmic threads, which is roughly
indicated in a rude model which I have constructed (textfig. 1). It is an indiarubber
balloon, with a band applied round the equator, to which threads are attached. The
threads are brought out through tubes, the inner orifices of which are carried some
distance into the interior. When the balloon is inflated through one of the tubes with
the threads loose, the result is such as represented in fig. 1 ; when they are drawn tight
* ZellenStudien, Heft 2, 1888. t Arch. f. Entwickelungsmek., Bd. v.
302
DR THOMAS H. BRYCE ON
up, on the other hand, the balloon is divided into two (fig. 2). This simple model is
not required, of course, to prove that such a system of threads, if contractile, or under
elastic tension, and attached to a cell membrane at the equator, will produce, or at any
rate initiate, cell division. Heidenhain's or Rhumbler's models show this quite well,
but the device described imitates in this one respect, I think, even better what I
believe actually occurs in this special case.
There is no apparent sign of growth of the cell membrane at the equator, which
is one of Rhumbler's secondary factors. When once the furrow is produced it quickly
completes itself, because the external pressure is now related to the two centres, and
division takes place in the neutral zone between them. That the subequatorial threads
Fig. 1.
Fig. 2.
become stretched out in the axial line between the centres is seen in the photograph
PL V. fig. 40, which closely resembles text, fig. 2 representing the model.
That the protoplasm has considerable ductility seems to be indicated by the tardy
return to the reticular or alveolar condition, and also by the drawing out of the spindle
remnant between the daughter cells into a thread of some length.
Turning for a moment to alternative hypotheses as to the structure of the corpuscles,
I think the idea that the phenomena are to be attributed to lines of strain in a homo
geneous and continuous colloid substance may be put aside. Although the alveolar
theory of Butschli is excluded in the strict sense of the term by the size of the alveoli,
the protoplasmic framework behaves much as the hyaloplasmic framework does in
Rhumbler's theory and the elastic framework in his model.
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 303
It would be beyond the scope of this paper to enter on the possible theories as to
the changes underlying the retractive phenomena in the protoplasm, but it may be noted
that as the centrosome does not enlarge during mitosis, there can be no actual centri
petal movement of the protoplasm on to that body. The same difficulty presents itself
if we supposed, with Rhumbler, that the centrosome acts by the abstraction of water
from the hyaloplasmic framework, causing it to thicken and condense, unless it were
further supposed that the water entered into new combinations in the centrosome, which
it is not very easy to accept.
The account I have given above is an attempt to explain merely the phenomena as
they are presented in the individual case, and does not involve a general theory of the
mechanism of mitosis. It seems at first sight radically different from that given by
various observers (Wilson, * Teichmann,+ and others) of the appearances in dividing
ova, in which the radiations are conceived as manifestations of an actual centripetal move
ment of the hyaloplasm. It may, however, be that the contradiction is one of appear
ance only. The essential factor is the same in both cases — a centripetal condensation of
the hyaloplasm. In very fluid protoplasm like that of the ovum, there may well be an
actual centripetal movement ; but in very viscous protoplasm like that of the red
corpuscles, which are undoubtedly firm and elastic bodies, the condensation may
involve only retraction without a flowing movement. If the framework is fixed peri
pherally the retraction would involve increased tension and the rays would become
contractile fibrils. Thus no one explanation will apply to all cases ; for if the centrosome
and its radiations are the expression of a condensation of the active protoplasm, due to
chemical or physical causes, the mechanical results will vary with the consistency of
the medium in which such condensation occurs.
II. Structure of the Leucocytes.
Though it is now well known that in all classes of vertebrates the blood of the adult
contains leucocytes of several different varieties, showing very different reactions to
various dyes, little is known about the first appearance of the white elements in the
blood of the embryo. The stage of embryonic life at which they appear seems to vary.
In Lepidosiren the blood is already at a very early stage provided with several different
kinds of leucocytes, but in the present writing I shall describe merely the morphology
of the different kinds of free cells I have found in the blood and tissues of the embryo,
reserving for a future communication the questions regarding the origin of the different
varieties, and the interrelation between them.
(1) Small Mononuclear Hyaline Corpuscles.
This form occurs sparsely in the blood, but more abundantly in the spaces adjoining
the posterior cardinal sinus.
* Arch. f. Entwickelungsmek., Bd. xiii. t Ibid., Bd. xvi.
304 DR THOMAS H. BRYCE ON
It measures 14 to 16 microms in diameter, and possesses a small halo of very delicate
protoplasm, which varies in amount from a zone hardly to be made out except under a
high power, to a welldefined envelope to the nucleus (fig. 23, PI. III. ; fig. 44, PI. V.).
The protoplasm is nongranular, is hyaline in appearance, and even under a magnifica
tion of 1500 diameters it is not possible to make out more than the vaguest suggestion
of reticular formation. In methylene blue and eosin preparations it is very delicately
stained by the basic dye, while in those tinted with Ehrlich's ' triacid ' mixture it has a
faint grey tinge.
I cannot with certainty demonstrate a centrosome. The nucleus is round, with a
coarsish chromatin reticulum, loosely arranged. It colours violet with methylene blue
and eosin, and no part of the nucleus is oxyphil, the linin taking a cold blue tint, while
the karyosomes are deep violet.
There is some doubt whether in all instances the nucleus is round, or whether there
is a notching at one pole. I have observed some such nuclei, and it is obvious that the
notching could only be seen if the section passed through a plane at right angles to it,
and through the centre of the body.
(2) Large Mononuclear Hyaline Corpuscles.
This variety occurs more frequently than the last, and is the commonest form seen
in the blood stream. It measures 24 to 26 fj. in diameter. The protoplasm varies in
amount, but is always merely a narrow zone surrounding the nucleus.
In methylene blue and eosin preparations it has a very delicate blue tint, and
high magnification reveals a very delicate meshwork, with microsomes at the nodal
points, which stain brightly with the blue dye (PI. III. fig. 24 ; PL V. fig. 45).
The nucleus is spherical ; the chromatin network is very loosely arranged, and therefore
in a section (fig. 24) one sees only rounded bodies with delicate threads radiating
from them. These are not true nucleoli or plasmosomes, but karyosomes. So far as
I can discover, plasmosomes do not occur in these embryonic nuclei. The staining
reactions are interesting. In iron haematoxylin and eosin preparations the karyosomes
are black and the general network red, but the chromatin parts more readily with the
black stain than the chromatin of the red corpuscles, so that in sections which are suit
able for a study of the latter the white cells are almost purely red. In methylene blue
and eosin preparations the karyosomes are deep violet and the network takes a blue
shade, but, as in the small corpuscle, there are no purely oxyphil granules. The deep
violet blue stands out in strong contrast to the delicate pure methylene blue staining
of the protoplasm.
In triacid material (fig. 25, PL III.) the network is green, and the karyosomes almost
invariably retain some of the acid dye. The colour is sharply distinguished from the
golden yellow of the chromatin of the red corpuscles, but also from that of the general
mesenchyme nuclei, which stain pure green in preparations which show a yellow tinge
in the leucocytes. In cells which show this staining, the chromosomes during division
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 305
have the same yellow tint as the karyosomes, showing that the material for the
chromosomes is at any rate chiefly drawn from them.
Though in the great majority of sections the nucleus appears spherical, it is more
than probable that corpuscles showing characters identical in other respects, but with
a notch at one pole, as represented in fig. 25, are merely cells cut in a plane at right
angles to that in which the rounded nucleated cells are cut. I have observed, however,
all degrees of notching, from a slight bay to an angular depression, such as seen in
fig. 25, or even to a linear fissure reaching to the centre of the nucleus, the two
walls of which are in close contact. In sections such as that drawn in fig. 25,
PL III. there is found lying opposite the notch a very imperfectly developed
centrosome. I have drawn it as a nearly circular darker portion of the protoplasm,
which is the ' attraction sphere,' staining like the protoplasm a neutral tint with
triacid. At the centre is a slightly darker circular spot, which I take for the
centrosome, but it is impossible to make out either a radial structure in the ' sphere '
or rays extending from it into the surrounding protoplasm. In iron hsematoxylin
and eosin preparations the same spot in the cell comes out as a homogeneous area,
staining of a darker red tint than the rest of the cytoplasm, but I have never seen
a darker spot in its centre.
There is no doubt that we have here to do with a protoplasmic area, which
corresponds to the area to be described in the next variety of leucocyte, showing an
active and operative centrosome, with its attraction sphere and rays.
In triacid preparations the protoplasm stains of a somewhat indefinite neutral
tint, and no granules are ever to be made out.
The cell represented in fig. 25, PL III. is certainly a leucoblast, but there is some
reason to believe that certain of the cells like that figured in fig. 24, PL III. bear a
relation to the Erythroblasts. These are cells of the same dimensions, but with
larger karyosomes and a coarser intervening network, and showing a concentric fibrilla
tion of the basiphil protoplasm. They will be dealt with in Part II.
(3) Polymorphonuclear Corpuscles.
This variety I have named in keeping with the general terminology of blood histol
ogy, on account of the lobed form of the nucleus. This body may, however, have many
forms (as seen in figs. 19, 20, PL II. ; 2729, PL III. ; and photographs 46, 47, 48,
PL V.). Sometimes, as in fig. 27, PL III., the superficial appearance is that of a
multinuclear corpuscle, but in reality the nucleus of that cell was single, but much
lobulated.
This group of corpuscles is characterised by the possession of a wellmarked
centrosome in active operation. They are frequently seen in active diapedesis.
Further, they always show, or almost always show, granules in their protoplasm. I
shall first describe the centrosome. In fig. 19, which is the same cell photographed
in fig. 49, I have drawn the body without filling in the granular cytoplasm.
306 DR THOMAS H. BRYCE ON
The centrosome is a large body, which stains a delicate grey in iron hematoxylin
preparations, many degrees lighter in tint than the intensely black centrosome of
the dividing erythrocytes. In sections counterstained with eosin the body is red.
It does not stain, therefore, like the chromatin. In methylene blue and eosin
preparations it is very faint, coming only very indistinctly out as a slightly darker
area on the faintly bluishred protoplasm. In triacid preparations it is very distinct
(fig. 31, PL III., and fig. 48, PL V.), and has a neutral tint. I have never seen in my
material any finer structure within the centrosome, nor can I make out any granules,
single, double, or multiple. I have therefore not adopted the term microcentum
(Heidenhain), # but have used the word centrosome in Boveri's sense.
Pound the centrosome there is a sphere distinctly radiate (as Van Beneden,
Heidenhain, and others have described), bounded by a circle of microsomes. This
separates the central from an astral zone, into which the fibres of the central sphere
pass. There are no outer circles of microsomes concentrically arranged, as described
for some cells (Druner). The radii are at first straight, then becoming vavy, they
seem to branch and join the general meshwork (fig. 20, PL II.). t
In fig. 21, PL II. is represented an amoeboid leucocyte. The extended pseudo
podium is not straight, but wavy, and its axis is occupied by a core of seeming fibrihse
passing from the centrosome. The cytoplasm around this central core shows an
exceedingly delicate meshwork structure, but it is difficult to be quite sure of this
towards the end of the pseudopodium, which is not well defined, and composed of
very delicate substance. From the centrosome to the sides of the nucleus there
seems to pass a core of seeming fibrillse, represented in the section by the two lateral
strands figured.
Another wandering cell is drawn in fig. 26, PL IV. It is passing through a
space between a number of other mesenchyme cells. The manner in which the
polymorphic nucleus is doubled up is interesting. The meshwork and radiations as
well as the centrosome are very obscurely revealed in these granular cells by methylene
blue and eosin, but it could be made out that at the extremity of the body the mesh
work was drawn out, and that delicate radiations from the centrosome passed into it.
Fig. 22, PL II. represents a corpuscle in which the centrosome has divided. The
' attraction sphere ' has apparently enlarged, and is not now bounded by a circle of
microsomes. Round the two centrosomes new radiations are developing within the old
sphere, which has now the appearance of an extremely fine feltwork, so fine that it is
difficult to convince oneself that there is any structure at all.
The protoplasm of these leucocytes is basophil when free from granules ; when
only a few granules are present, it stains a delicate warm blue with methylene
blue and eosin, and shows a very delicate faint meshwork. The granules
colour intensely with eosin, are copper red after treatment with triacid, and
blacken after iron hematoxylin. They vary in size in the same cell and in
* Archivf. mikr. Anat., Bil. xliii. t Gf. Gulland, Jour, of Phys., vol, xix.
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOS1REN PARADOXA. 307
number in different cells. In many they are scattered very sparingly through the
protoplasm (fig. 27, PL III.), in others they are closely packed (fig. 29, PI. III.), while in
a smaller number the whole cell body is a uniform mass of closely apposed large granules
of nearly uniform size (fig. 30, PL III.). These lastdescribed corpuscles may be
classified as
(4) Eosinophil Leucocytes.
Though they have all the appearances of eosinophils described in adult animals, it
is not contended that they necessarily belong to the same category.
In leucocytes in which the granules are sparingly distributed, their dimensions vary
from the minutest particle to bodies 2 to 4 m in diameter. They are highly refractile,
and this gives rise to some difficulty in determining their relations to the meshwork.
In the drawings I have represented them as seen surrounded by their halo of refraction.
Apart from their actual relationships to the alveoli of the meshwork, it is quite
certain they are not the nodal points of a reticulum, but are clearly metaplastic,
probably minute drops in the protoplasm, which run together to form larger granules.
Exactly similar granules are found in the yolk cells, so that it is probable that these
in the leucocytes are derived from that source, in which case they could be neither
secretory nor excretory products ; but to this question I shall return in the second part
of this memoir.
There is always a space clear of granules round the centrosome (figs. 29 and 31,
PL III.).
The nucleus varies greatly in shape, but in its other characters it agrees with that
of the mononuclear leucocytes. The simplest form in which it is found is the horse
shoe shape (PL II. fig. 19 ; PL V. fig. 49). Between this and the complicated lobed
condition there are all varieties. Sometimes it is ringshaped (PL V. fig. 47), while
sometimes it is formed of quite a number of lobes (PL II. fig. 20 ; PL III. fig. 27).
In no case are the lobes detached from one another to produce a multinucleated cell.
They are always joined by attenuated portions of the nucleus. The leucocyte figured
in fig. 27, PL III. has all the appearance of a polynuclear cell, but careful scrutiny
proved that the several lobes were connected together like the two lobes of the nucleus
seen in fig. 29 on the same plate.
I have found all the varieties of the leucocytes in mitotic phases, but these are few
in number at the stages examined compared with the dividing erythrocytes. The
polymorphonuclear may divide with the nucleus in the horseshoe form. The general
character of the karyokinetic phases is the same as in the erythrocytes, but the chromo
somes arising from a relatively loose and scantier chromatin reticulum are much finer
and smaller, and the achromatic structures are of great delicacy. It is interesting to
note that, as has been observed in other cases, the centrosome which is so large in the
resting phases is reduced during mitosis to a very fine granule, hardly to be demonstrated.
As the object of the present writing is more descriptive than theoretical, I do not
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 11). 46
308 DR THOMAS H. BRYCE ON
propose to enter into a detailed discussion of all the theoretical points on which my
observations bear. The problems relating to the histogenesis of the erythrocytes, and
the origin and interrelations of the leucocytes, will be dealt with in the second part of
the memoir. Here I shall only deal briefly with certain more strictly cytological con
siderations, and refer to certain conclusions which the contrasting characters of the
erythrocytes and leucocytes seem to warrant.
It is to be noted that as the close juxtaposition of the blood elements has exposed
them to identical conditions both of fixation and staining, the different pictures pre
sented afford a secure ground for comparative study.
In the matter of the protoplasm, the material affords examples of all grades between
a purely structureless condition and a coarse reticular formation. I have already
sufficiently discussed the reticular structure of the erythrocytes, and expressed my
belief that it represents an actual disposition of the protoplasm in the living cell.
During mitosis it has been demonstrated that this framework is converted into the
achromatic figure. In the leucocytes which show a definite but extremely delicate
reticulum, this plays the same part in mitosis as the large network of the erythrocytes.
The appearances differ in degree only, not in kind, and therefore there is no sharp
line to be drawn between the two, whether one accepts the reticular or the alveolar
hypothesis. It would seem to be more or less a matter of the relative proportion of
active and passive elements, and of variations in consistency.
The differential staining of the chromatin of the erythrocytes is a point of suggestive
interest, but I am not competent to deal with the questions of cell chemistry involved.
While the behaviour to iron hematoxylin might be due merely to differing physical
properties, the reaction to the other dyes indicates a chemical differentiation of the
chromatin which must be in some way connected with the functions of the corpuscles.
The most suggestive of the contrasts observed is that in the characters of the
centrosomes.
In the erythrocytes, which are passive bodies, in the resting stages there is no
centrososome discernible. The body is related only to the mitotic phenomena, and
when the kinetic phase is past it disappears as such. Every fact in its history points
to its being merely the central point of a cytoplasmic condensation, whatever may be
the physical or chemical changes involved.
In the leucocytes the centrosome always stains with the cytoplasm. In its full
panoply of sphere and aster it is only seen in the leucocytes which undergo amoeboid
movements. This fact supports the view that it is related to these movements, and
this is actually demonstrated by leucocytes caught in amoeboid movement (fig. 21,
PI. II.). That some cytoplasmic activity exists, centred on the centrosome, is clear,
but what the nature of the activity may be is another matter.
My observations are too few to warrant my going into this question. One would
require to see many more amoeboid leucocytes at all stages than I have done to form
any opinion on the general question. I put forward the facts I have observed merely
THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 309
in support of the supposition, that whatever may be the explanation of the outputting
of the pseudopodium in the case of the leucocytes, the withdrawal and forward move
ment must be referred to the centrosome, with its sphere and radiations ; and so far as
the appearances go, they are in favour of the idea that the centrosome in this case also
is the centre of a contractive force.
In the conversion of the leucoblast into the leucocyte, the attraction sphere and
aster are gradually unfolded as the cell body increases in size. The nucleus is at first
central, bub later assumes an eccentric position, while the sphere moves towards the
centre of the cell. I would explain this in the same terms as I explained the rounding
up of the erythrocyte, and the separation of its centrosomes. The centrosome being
here single, however, it comes to a position of equipoise in the centre of the protoplasm.
My reading of the structure of the leucocyte is different from that of Heidenhain,*
in so far as the radii seem to me to branch and join the general reticulum, which I
believe (with the necessary reservations) probably represents an alveolar disposition of
the protoplasm, but they act quite like his organic radii, in respect of the movement
of the sphere. With regard to the form of the nucleus in relation to the movement of
the sphere, my observations, so far as they go, seem to agree with his in matter of fact,
but I have not followed out the point in such detail as to follow him into the domain of
theory.
EXPLANATION OF PLATES.
The drawings were done, in everything but the very minute detail, by aid of the camera lucida (Abb^).
The lenses employed were the 3 mm. and the 2 mm., both L4 numerical aperture, apochromatic
objectives of Zeiss, combined with the compensating oculars 8 or 12. The magnification indicated was
ascertained by the stage micrometer. It is rather greater than the magnification given by the com
binations used, the excess depending on the depression of inclined drawingtable beyond the visual distance.
The coloured drawings were tinted with the same stains as used for staining the sections. The watery
solutions of aniline dyes colour smooth Bristol board very delicately, and permit of a degree of verisimilitude
difficult to attain with ordinary watercolours.
The photographs were all taken with the 3 mm. 1*4 numerical aperture achromatic objective and No.
4 projection eyepiece, at a distance which gave a magnification of 800, with the exception of fig. 37, in
which the magnification is 1500.
Plate I.
Fig. 1. Section red blood corpuscle in plane parallel to surface of disc, x 1200 d. Compare photo
graph, PI. IV. fig. 32.
Fig. 2. Section of same in place at right angles to surface of disc, x ] 200 d. Compare photograph,
PI. IV. fig. 33.
Fig. 3. First stage of mitosis. Section passes through corpuscle in a plane vertical to its flat face. It
is rounding up for division. Possible phase of single centrosome placed in a projection which has risen from
centre of disc, x 1200 d.
Fig. 4. Similar stage in larger corpuscle. Two centrosomes. x 1 200 d.
Fig. 5. Corpuscle with two independent centrosomes. x 1200 d.
Fig. 6. Corpuscle with two centrosomes which have appeared separately as in last, x 800 d.
* Loc. cit.
3 1 THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA.
Fig. 7. Case of want of synchronism between centrosomal and nuclear cycles. Polar view of corpuscle
with nucleus in stage of spireme, while centrosomes still close together x 800 d.
Fig. 8. Corpuscle which has rounded up for division. Nucleus in tight spireme stage. x 1200 d.
Fig. 9. Drawing apart of the centrosomes. Chromatin thread divided into Vshaped loops, each of
which has a long taillike process from one of its limbs, x 800 d.
Fig. 10. Same, a little later, x 2000 d. Compare photograph, PI. IV. fig. 36.
Fig. 11. Stage in rotation of spindle system, x 2000 d. Compare photograph, PI. IV. fig. 37.
Fig. 12. Fully developed metaphase. x 2000 d.
Plate II.
Figs. 13, 14, 15. Three stages in anaphase, illustrating especially the emergence of the ' subequatorial
fibres.' x 2000 d. With fig. 15 compare fig. 39, PI. IV.
Figs. 16, 17, 18. Telophases illustrating division of cell body ; formation of intercellular spindle rem
nant ; reduction of same to form midbody, which disappears by being drawn out into a thread of some
length ; disappearance of centrosome in situ without previous division ; condensation of chromosomes to
form a continuous and homogeneous mass. Fig. 16 x 2000 d. ; figs. 18, 19, 1200 d.
Fig. 19. One variety of polymorphonuclear leucocyte, showing large centrosome, radiate sphere, and
single circle of microsomes, x 1500 d.
Fig. 20. Same, with ringshaped and lobed nucleus. The cytoplasmic meshwork indicated. There
are a few minute granules, x 1500 d.
Fig. 21. Amoeboid leucocyte, x 2000 d.
Fig. 22. Polymorphonuclear leucocyte, showing division of centrosome within old sphere, x 1800 d.
Plate III.
Fig. 23. Small mononuclear hyaline leucocyte, methylene blue and eosin. x 2000 d.
Fig. 24. Large mononuclear cell, methylene blue and eosin. x 2000 d.
Fig. 25. Large mononuclear leucocyte, with notched nucleus and centrosome ; Triacid. x 2000 d.
Fig. 26. Polymorphonuclear leucocyte ; amoeboid, x 1500 d.
Fig. 27. Another form of same, with muchlobed nucleus and scattered small granules. Centrosome
not in section. Methylene blue and eosin. x 1500 d.
Fig. 28. Another form of same, showing nature of granulation in protoplasm. Centrosome not in
section, x 2000 d.
Fig. 29. Same, with closely packed granules, x 2000 d.
Fig. 30. A further stage of same, showing all the characters of an eosinophil leucocyte, x 2000 d.
Fig. 31. Same, showing reaction to triacid. x 2000 d.
Plates IV. and V.
[All the photographs were taken at a magnification of 800 d.]
Figs. 32 and 33. Resting red blood corpuscles.
Figs. 34 to 43. Sequence of mitotic phases. Fig. 37 x 1500 d.
Fig. 44. Small mononuclear leucocyte.
Fig. 45. Large mononuclear leucocyte.
Fig. 46. Polymorphonuclear leucocyte, with centrosome and sphere.
Fig. 47. Same, with ringshaped nucleus.
Fig. 48. Same, with fine granules.
Fig. 49. Same, with large closelypacked granules, i.e. eosinophil leucocyte.
The photographs in reproduction have lost much of the delicacy of detail seen in the gelatinochloride
prints.
'logy of the Blood of the Larva of Lepidosiren Paradoxa Part I P]
s Roy. S<
"
•■•■
■
£ _ . ±
■
[
&«% fit
10
12
Bryce Histology of the Blood of the Larva of Lepidosiren Paradoxa Part I Plate n.
Roy So
XLI.
16
17
18
15
'
21
19
20
22
Bryce iogy of the Blood of the Larva of Lepidosiren Paradoxa Part I Plate
III.
iTrani Soc Edm r
Vol.XLI.
23
24
25
26
27
28
• ".•;•■■•
29
30
31
WemerAWmtei lit!
Trans Rov. Soc, Edin.
Vol. XLL, Plate IV
BRYCE: HISTOLOGY OF THE BLOOD OF THE LARVA OF LEPIDOSIREN PARADOXA
PART I.
FIG. 32.
'"'■■
FIG. 33.
w.
FIG. 34.
^^^
Sfc
'
I
FIG. 35.
f
ma,
FIG. 38.
FIG. 36.
FIG. 39.
FIG. 37.
*
V
FIG. 40.
%&AL
Trak, Roy. Soc, Edin.
Vol. XLL, Plate V.
BRYCE: HISTOLOGY OF THE BLOOD OF THE LARVA OF LEPIDOSIREN PARADOXA
PART I,
m
$
FIG. 41.
%
<t »
■
m
FIG. 44.
I
J
FIG. 47.
*i
r >
»>■■•
FIG. 42.
I
»
' ,
*
FIG. 45.
v
FIG. 48.
**
FIG. 46.
*
FIG. 49.
( 311 )
XII. — The Action of Chloroform upon the Heart and Arteries. By E. A. Schafer,
F.R.S., and H. J. Scharlieb, M.D., C.M.Gr. (From the Physiological Laboratory
of the University of Edinburgh.)
(Communicated March 21, 1904. MS. received August 17, 1904. Issued separately December 14, 1904.)
The original design of this research was to determine whether the extract of
suprarenal medulla (or its active principle) has the power of antagonising the effects
of an overdose of chloroform upon the heart and arterial system. Incidentally the
research became extended so as to cover the action of certain other antagonising
agents. It further appeared necessary, as the investigation proceeded, to subject the
action of chloroform upon the vascular system to renewed study. For although, as
the result of numerous recent researches, physiologists are in agreement regarding the
general effect of the drug upon the heart, there yet remain various points requiring
elucidation both as regards its effect on the heart and on the arteries.
Effect of Chloroform upon the Arteries.
Singularly little is precisely known as to the effect of the drug upon the arterial
system. The most generally received opinion has been that adopted by Bowditch and
Minot * to the effect that chloroform exerts, besides a specific action on the heart, a
paralysing influence upon the whole vasomotor system, and that the fall of blood
pressure which accompanies its administration is due as well to the dilatation of
vessels as to the effect which it produces upon the cardiac musculature. On the other
hand, Arloing,! as the result of observations on the rate of flow through the carotid,
made by means of the hsemadromograph, inferred that a constriction of arterioles is
produced by the drug. Dastre J came to the same conclusion, and referred to it the
pallor of the face which is seen in chloroform administration. But it is obvious that
a diminution of rate in the carotid might be caused by dilatation of vessels in the
splanchnic area, so that these observations cannot be regarded as conclusive. Gaskell
and Shore,§ in their cross circulation experiments, obtained distinct evidence of stimula
tion of the vasomotor centre ; constriction of arterioles and rise of bloodpressure
occurring as the result of the action of the drug upon the medulla oblongata. Roy
and Sherrington  inferred that constriction of cerebral vessels is produced by chloro
* Boston Med. and Surg. Jour., 1874. Cf. Leonard Hill, Brit. Med. Jour., April 1897.
t Thkse, Paris, 1879. J Les Anesthetiques, 1890.
§ Brit. Med. Jour., 1893, vol. i. \\Jour. Physiol., vol. xi. p. 97, 1890.
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 12). 47
312 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
form, while Hurthle* found evidence of dilatation of these vessels, followed by
constriction. Newman t observed constriction of pulmonary capillaries in the frog as
the result of chloroform inhalation. Sherrington and Sowton, working with the
isolated mammalian heart perfused with Ringer's solution by Langendorff's method,
observed a diminished flow of the perfusion fluid when chloroform was added to it ;
this they were inclined to ascribe to a contraction of coronary vessels under its influence.
C. J. Martin § has suggested that this diminution of flow through the coronary
vessels may be accounted for, without the necessity of assuming constriction of those
vessels, by the fact that a diminished action of the cardiac musculature, such as chloro
form produces, may tend by itself to diminish the rate of circulation through its vessels.
To this we may add that in a heart which is separated from its surroundings and fed
by the perfusion of fluid under pressure into the root of the aorta, in which therefore
the mechanical conditions are very different from those which obtain normally, the
aortic valves do not necessarily act efficiently, but often permit of some passage of fluid
into the cavity of the left ventricle, and through this into the left auricle, and so out
by the cut pulmonary veins ; and the extent of this valvular defect with the consequent
leakage will vary with the condition of tone of the heart and the force of its contractions.
Opinions on this subject being thus divided, it appeared important in the first
instance to determine what is precisely the action of chloroform upon the arterial
system. The method which we have used for this purpose is the classical one of
perfusing the vessels with blood or saline fluid containing the drug in solution. The
chloroform used for this purpose and in most of our experiments has been Duncan &
Flockhart's, sp.gr. 1*49. The result of our preliminary experiments  showed that a
solution containing from 1 gramme to 5 grammes of chloroform to the litre of circu
lating fluid produces a marked constriction of the frog's arterioles, and that this con
striction is apparent whether the medulla oblongata and spinal cord are left intact or
destroyed. These observations established the fact that for high percentages of chloro
form (5 grammes per litre is approximately a saturated solution, and 1 gramme per litre
is therefore onefifth saturated) there is a pronounced excitation by the drug of the
musculature of the arterioles — whether operating directly or through the vasomotor
nerveendings, our preliminary experiments did not decide — which may contract under
its influence to such an extent as almost to arrest the flow of circulating fluid.
Since the publication of these preliminary results, C. J. Martin,§ in confirmation
of earlier experiments in conjunction with Embley,H has made observations upon the
mammalian kidney by the plethysmography method which appear to indicate that
in dilute solution — the actual dosage was not determined, but the perfusing fluid
(blood) was first passed through the lungs, into which a mixture of air and chloroform
vapour was pumped — chloroform has the effect upon the bloodvessels ascribed
* Pfliiyer's Arch., vol. xliv. p. 596, 1889. t Jour. Anat. and Phys. vol. xiv., 1879, p. 495.
X ThompsonYates Laboratories Reports, 1903, vol. v. § Private communication.
 Communication to the Physiol. Soc. ; Jour. Phys., vol. xxix., 1903.
I Brit. Med. Jour., April 1902. lancet, 1902.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 313
to it by Bowditch and Minot, viz., that of producing dilatation. In repeating and
extending our experiments we have therefore included the effect of the perfusion of
mammalian organs with various strengths of chloroform dissolved in Ringer's solution.
Perfusion of Frog Vessels ivith Chloroform dissolved in Ringer's Solution.
The Ringer's fluid used consisted of NaCl, 6 grammes ; CaCl 2 , 0*1 gramme ; KC1,
0*075 gramme ; NaHC0 3 , 0*1 gramme per litre. The chloroform was either dissolved in
this solution in proportion determined by weight, or some of the fluid was saturated by
being shaken up with and kept over an excess of chloroform, and was assumed to con
tain 1 part chloroform to 200 Ringer, this being the amount water will take up at the
ordinary temperature of the air (15° C). This saturated solution was mixed with vary
ing proportions of normal Ringer. A fine cannula having been tied into the bulbus
aortse of the frog (R. esculenta or R. temporaria), the fluid was allowed to pass by
gravity, at a pressure varying in different experiments from 50 mm. to 150 mm. of
water, through the vascular system, and to drip from the extremities of the toes. In
our earlier experiments the mode of determining the rate of flow was to count the
number of drops per minute ; but this method, although serving to indicate any differ
ences of vascular calibre which are marked, is not sufficiently accurate for slight varia
tions, since the size of the drops is liable to vary somewhat with differences of surface
tension of the fluid, and the amount of dissolved chloroform or of intermixed blood and
lymph may affect its surface tension. In all later experiments, therefore, the amount
of fluid perfusing in a given time was accurately measured. Only the results thus
obtained are included in this communication.
The result of these perfusion experiments with Ringer's fluid containing dissolved
chloroform may be shortly stated as follows : —
With the strongest solutions, i.e., from saturated (1 in 200) down to solutions con
taining 1 in 500, a very marked constriction of the arterioles is the result of perfusing
with chloroformRinger, so that the flow of the perfusing fluid becomes very slow, and
may almost cease. With increase of dilution the amount of constriction, as registered
by rate of flow, becomes less ; but although very slight when the dilution is consider
able, we have been able to substantiate constriction with solutions as weak as 1 in
20,000. On the other hand, no solution of any strength when perfused through the
frog's vessels has given evidence of dilatation of arterioles, the weaker solutions having
simply shown themselves inert. If for the chloroform Ringer which has been passed
for some minutes through the vessels, and has produced the diminutions of flow above
indicated, normal Ringer be now substituted, the flow again becomes more rapid, but
the original rate is rarely again obtained ; in fact, after the chloroform solution has
been in action for some minutes, even if the strength of the solution be such as to
be insufficient to cause actual constriction of arterioles, there is a tendency towards a
gradual diminution in the rate of flow, which appears to be caused by oedema of
314 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
the tissues ; the effect of the chloroform solution on the endothelium of the vessels
being such as to render the capillaries more permeable to saline solution.
The following experiment will serve as an example of the results obtained with weak solutions of
chloroform in Ringer's solution. The numbers represent the amount flowing through the vessels in equal
periods of time.
Immediately before chloroform perfusion ......... 28 "5 c.c.
During perfusion of fluid containing 1 part chloroform in 20,000
Immediately after perfusion of chloroformRinger . . .1st period
2nd „
3rd ,,
225
175
21
25
Perfusion of Mammalian Vessels ivith Chloroform dissolved in Ringer s Fluid.
The kidney, leg, and heart of the cat, rabbit, or dog were employed, and the method
of perfusion was the same as for the frog, except that the head of pressure was higher
(80 to 100 mm. Hg.). The Ringer solution had the composition: NaCl, 9 grammes;
CaCl 2 , 0"24 gramme; KC1, 0*42 gramme; NaHCO s , 0"1 gramme; distilled water,
1 litre, and was warmed to 38° C. by being passed through a glass spiral contained in
a waterbath before being conducted to the organ to be perfused, which was itself also
kept in a warm chamber at the same temperature. The perfused fluid was either
collected in a graduated measure and the amount flowing in a given time recorded, or
it was caused to work an automatic " filter," so arranged that every 7 c.c. of fluid
produced a seesaw of the filter, and this was recorded by a magnetic signal. In some
experiments Ringer's solution, containing a known percentage of chloroform, was, after
the normal record of flow had been obtained, allowed to pass for a certain time through
the vessels in place of the ordinary Ringer, and was then again replaced by ordinary
Ringer, the rate being recorded before, during, and after the passage. In other
experiments a chloroformRinger of known strength was injected by a fine hypodermic
needle through the indiarubber supplytube of the perfusion apparatus, so as to mix
with the inflowing normal Ringer. The amount of dilution of the chloroformRinger
so perfused was calculated from the amount of fluid flowing through the kidney during
the actual time occupied by the injection. This method has the advantages (l) that
the chloroform solution only acts for a short time upon the kidney vessels, and is less
liable to cause a permanently deleterious effect ; and (2) that the conditions of flow are
maintained the same throughout, for if the injection is performed very gradually, no
perceptible increase of pressure is caused by it. (It is scarcely possible to change the
flow from one vessel to another, as in the ordinary method of testing perfusion, without
causing a temporary effect of some kind upon the pressure of the perfusing fluid.)
The results yielded by these methods show that in mammalian as in frog's vessels
the effect of chloroform solutions of a certain strength is to cause marked constriction of
the arterioles and consequent diminution in the rate of flow of the perfusing liquid. If
the flow lasts for a short time only, the rate is soon recovered, but prolonged perfusion
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 315
— even with (socalled) normal Ringer, and much more with Ringer containing chloro
form — is accompanied by oedema and a consequent permanent diminution in rate
of flow.
Kidney. — The strengths of solution which produce constriction in the vessels of the
kidney are from saturated (1 in 200) to 1 in 1000 (perhaps somewhat more dilute).
But whereas, in the entire frog, solutions weaker than those which cause constriction do
not produce dilatation, and have no apparent effect apart from the gradual oedema which
results from prolonged action, in the mammalian kidney the effect of weaker solutions
is, as C. J. Martin has stated, to cause dilatation. This result has been obtained with
solutions of from 1 to 1500 to 1 in 20,000 (in one instance); weaker solutions gave no
result.
Coronary Vessels. — In employing the heart we have always taken the precaution of
tying the pulmonary veins, to prevent loss of circulating fluid by regurgitation past the
aortic and mitral valves. The effect of chloroform upon the coronary arteries is to produce
constriction in all strengths from saturated to 1 in 10,000. The stronger solutions of
chloroform cause so marked a diminution of the rate through the coronary vessels as to
almost arrest the flow of fluid ; and this is not due to arrested cardiac action, for
on substituting normal Ringer for chloroformRinger the rate of flow returns to normal
long before the action of the heart recommences. With weaker solutions the effect is
also to produce diminished flow, and at no condition of dilution have we obtained
evidence of dilatation of vessels.
Limbs. — For this purpose the hind limbs of the rabbit and cat have been used.
The results are precisely the same as in the case of the coronary arteries of the mammal
and the systemic arteries of the frog. Evidence of constriction has been obtained with
all strengths from 1 in 200 (which arrests the flow altogether) to 1 in 10,000 (which
causes a slight diminution). More dilute solutions are inactive ; we have obtained no
evidence of dilatation in these vessels.
The following may serve as examples of the results : —
Kidney of Rabbit :
Amount flowing before chloroform perfusion ........
,, „ during „ „ (1 in 1500 Ringer) ....
,, ,, after ,, ,, .... ...
The same Kidney, later :
Amount flowing before chloroform perfusion ........
during „ „ (1 in 700) . . . .
Kidneys of Kitten :
Amount flowing before chloroform perfusion ........
„ during „ „ (1 in 20,000)
, , ,, after ,, ,, ........
In this and the next experiment, as the increase was progressive and there was no return towards
normal, it is possible that the increase of rate may not have been due to the chloroform. But
it is clear that the drug has not caused constriction of the kidney vessels.
42
c.c.
49
>>
35
5 »
39
C.C
17
5 „
57
C.C.
60
!!
62
)J
316 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
The same Kidneys, later :
Amount flowing before chloroform perfusion 64 c.c.
„ during „ „ (1 in 5000) 67 „
„ during „ (1 in 2000) 70 „
Subsequent perfusion with suprarenal produced strong constriction, showing that the arterioles
were still active.
The effects of stronger and weaker solutions upon the kidney vessels is also well illustrated in the
accompanying tracings, which show (in fig. 1) the effect (marked constriction of vessels with rise of pressure
Fig. 1. — Perfusion of rabbit's kidney with Ringer's solution. At the time marked by the signal 4 c.c. of the same solution,
containing 0*5 per cent, of chloroform, and at the same temperature, was slowly injected into the supply tube. The
mixture of this solution with that passing through the tube at the time gave a fluid containing 0*2 per cent. (1 in 500)
actually perfused. Notice the rise of pressure due to constriction of the bloodvessels, followed after the passage of the
fluid by a dilatation ; also the great diminution in outflow, followed by a slight increase.
a, Register of mercury manometer ; b, movements of " seesaw," registered by air transmission : each up or down move
ment represents 7 c.c. of fluid discharged ; c, time in 10 sees. ; d, signal line and pressure abscissa.
Fig. 2. — Tracing similar to that shown in fig. 1, but with injection of 4 c.c. of 1 in 1000 chloroform, and actual perfusion
of 1 in 6000. Notice the fall of pressure and the increase in rate of discharge, indicating dilatation of arteries.
and great diminution of flow) of perfusing a solution of Ringer containing 1 in 500 chloroform ; and (in
fig. 2) the effect (dilatation of vessels with fall of manometric pressure, and increase of rate of flow) of passing
a solution containing 1 in 6000 through the kidney.
Heart of Cat ;
Amount flowing through coronary system before chloroform ..... 44 c.c.
„ „ „ „ „ during „ (1 in 10,000) . . . 415 .,
n n n ii i» illlCr ,, . 't't ..
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
317
Another Heart :
Amount flowing through coronary system before chloroform .....
„ during „ (1 in 4000) .
,, ,, „ ,, ,, alter ,, .....
Heart of Rabbit :
Amount flowing through coronary system before chloroform .....
„ „ during „ (1 in 1500) .
,, ,, ,, ,, ,, aitei ,, . ...
The same Heart, later :
Amount flowing through coronary system before chloroform .....
„ ' „ „ „ during „ (1 in 1000) . .
»» j» jj ? j jj alter ,, .....
The tracing shown in fig. 3 gives an illustration of the effect produced upon the heart by a still stronger
chloroform solution, the rate of rjerfusion falling during the passage of the chloroform from 21 c.c. per minute
to 5 c.c. per minute, and gradually recovering as the chloroform was washed away. It will be noticed that
the recovery of the vessels appears before the contractions of the heart reappear. It can also be seen that
the latency of the arterial contraction is longer than that of the heart paralysis which the chloroform produces.
45
c.c.
32
J3
42
>?
58
c.c.
40
?!
48
i>
48
C.C.
34
n
42
1 )
Fig. 3. — Effect of perfusing 20 c.e. of chloroformRinger ( = 1 in 500) through the coronary vessels of the rabbit.
a, tracing of manometer connected with supply cannula ; b, register of flow from coronary veins : each interval
represents 7 c.c. ; c, time in minutes ; d, signal marking period of injection into supply tube.
Note the diminution in rate of flow, and subsequent commencing recovery although the heart remains in a condition of
arrest. The rate of movement of the paper is too slow for the individual heartbeats to be seen on the manometer tracing.
Hind Limbs of Rabbit :
Amount flowing through limbs before chloroform .......
„ „ „ „ during ,, (1 in 10,000) . . . .
» )> !> >) alter ,, .......
Hind Limb of Rabbit :
Amount flowing through limb before chloroform .......
,, ,, ,, ,, after passage of 10 c.c. of 1 in 2000 chloroformRinger
,, ,, ,, „ in subsequent period ......
These observations show that the kidney differs from the other organs investigated
in the fact that the more dilute solutions of chloroform produce an increased flow
through the kidney vessels, whereas in the other organs (heart, limbs) the effect of the
drug is always in the direction of vasoconstriction. The difference is a remarkable one ;
but without discussing it at greater length, we may point out that dilatation of the
renal vessels is the normal response of the organ to all but a very few excitants, whereas
the normal response of most vessels to an excitation is contraction, and it is possible
therefore that the explanation is connected with this difference of " habit " of the
kidney vessels as compared with the systemic vessels generally.
86
c.c.
77
j?
84
»)
44
c.c.
39
)?
42
J)
318
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
The action of chloroform upon the vessels when perfused through the isolated organs
is a direct action upon the muscular tissue, and not, as in the case of suprarenal extract,
upon the terminal apparatus of the vasomotor nerves. This is shown by the fact that
apocodeine, which in sufficient dose abolishes the effect of adrenalin,* does not abolish
the effect of chloroform in producing vasoconstriction.
The following experiment may be quoted in illustration of this statement : —
Hind Limb of Rabbit. — Perfusion with 1 in 2000 chloroform reduced the rate of flow during each period of
time from 44 c.c. to 39 c.c. After recovery, perfusion with 0"0001 gramme hemisine (adrenalin) added
to the normal Ringer brought it down to 10 c.c. After recovery, perfusion with Ringer solution,
containing O'OOOl gramme hemisine and  0075 gramme apocodeine, caused only a slight reduction,
soon disappearing. After recovery, admixture with the perfusing fluid of 10 c.c. of chloroform
Ringer ( = 1 in 500), containing 002 gramme apocodeine, caused an almost complete arrest of flow
during several minutes. Fig. 4 is a graphic record of this experiment.
In all cases the drugs were injected into the tube which supplied the normal Ringer, and the
solution became mixed with a certain proportion of this, and warmed to the same temperature by
passing through the glass spiral before reaching the organ which was perfused.
As a further proof that chloroform acts upon the muscular tissue of the arterioles in
Fig. 4. — Effect of injecting 10 c.c. of chloroformRinger ( = 1 in 500) containing 0'02 gramme apocodeine through the vessels of
the hind limb of the rabbit.
a, b, c, d as in fig. 3. The tracing is taken on a more slowly moving surface than that in fig. 3. The initial pressure
of the perfusion fluid was lower in this experiment than in the experiment shown in fig. 3, and the supply less free : this,
as well as more complete constriction of the arterioles, accounts for the fact that the manometer tracing is much affected
in the one case and scarcely at all in the other.
producing contraction may be adduced the observation that its action can be got after
the neuromuscular endapparatus has lost its irritability. Thus in the kidney of a
rabbit, which had been killed three hours previously and in which the injection of
 0003 gramme hemisine (adrenalin) produced no effect whatever upon the rate of flow,
injection of 20 c.c. of 1 in 200 chloroformRinger into the supplytube reduced the rate
from 56 c.c. to 28 c.c.
All recent observers are agreed that the fall of bloodpressure which is caused by
chloroform is essentially due to its effect upon the heart muscle, the action of which
is weakened and eventually paralysed by the drug. Martin and Embley are, as we
have seen, inclined to ascribe the fall — in a minor degree — partly to the dilatation
which may be produced in the peripheral vessels by small doses of the drug. But
since Gaskell and Shore have shown that the effect of chloroform is to excite the
vasomotor centre in the medulla oblongata, and thus to cause contraction of the
* Dixon, Jour. Phys., vol. xxx., p. 97, 1904. Also Brodie and Dixon, ibid., p. 476.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
319
arterioles — an observation which Martin and Embley themselves confirm — it is
extremely improbable that, while this centre is in action, the dilating effect, if really
existent, upon the periphery would be at all apparent. On the other hand, with
stronger dosage of chloroform, the direct effect upon the arterioles is always one of
constriction. It follows, therefore, that at the beginning of a chloroform inhalation there
will be a tendency to counteract the fall of bloodpressure due to heart weakening, by
Fig. 5. — End of a fatal chloroform inhalation. Dog, 6570 g. Inhalation through trachea tube of air strongly charged with
chloroform vapour.
A, bloodpressure curve ; B, line 1 centimetre below zero of bloodpressure ; C, costal respiration (the small waves upon
this are heart movements) ; D, diaphragmatic respiration. Time in 10 sees. The signal marks the removal of chloroform.
Respirations ceased 20 sees, before the heart. The subterminal rise in bloodpressure which sometimes occurs is shown
in this tracing. The increase in size of the manometer excursions is due to a gradual slowing of the heart rhythm, and
does not represent an increase of force of the contractions.
excitation of the vasomotor centre, and later on, while this may still be active,* a similar
tendency to counteract the fall by direct excitation of the peripheral arterioles. As a
general rule, the action of the drug upon the heart is so marked as to more than
counterbalance the arterial constriction, however produced. But in certain cases a
* Reflex constriction of bloodvessels can be obtained, ou stimulating an afferent nerve, even if chloroform anaesthesia
is very pronounced, showing that even in deep anaesthesia the vasomotor centre is still active, although its activity is
no doubt lessened. Cf. Bowditch and Minot, op. cit. Further, chloroform does not diminish the excitability of the
peripheral vasomotor nerves (Scheinensson, Oentralbl.f. d. med. Wiss., 1869, p. 105).
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 12). 48
320
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
brief rise of pressure at the beginning of an experiment is sufficiently manifest, and in
others again there is apparent, even when the respirations are maintained, a rise towards
the end of the experiment. Since in these cases asphyxia is eliminated, and the heart
is probably not beating more but less strongly, such a subterminal rise can only be
ascribed to the excitatory action of the drug upon the vasomotor centre, or directly upon
the arterioles. It is exemplified in fig. 5, which shows the tracing from the latter half
of an acute chloroform poisoning, terminating by a slowing and arrest of the respirations,
followed after a few seconds, suddenly, by complete arrest of the heart. In this tracing
it will be observed that long before the failure of the respiration begins to show itself
there is a decided tendency to rise on the part of the arterial pressure, although the
heart at this time is not beating more but rather less strongly (the increase in size of
_rs_^V_rV_y
^WV.Y'.'.flWl\VY<WvW,,.,,r.~~.
Fig. 6. — Tracing (dog) showing marked secondary rise of bloodpressure during chloroform inhalation, probably due to early
failure of respiration. The chloroform was administered between the two marks on the signal line. Notice the cardiac
inhibition, which in this case is more gradually developed than usual, and the subsequent escape of the ventricle, which
continued to beat feebly for a minute or two, but with hardly any rise of bloodpressure. Artificial respiration by pump
commenced 10 minutes after natural respiration had ceased, failed to effect recovery.
a, bloodpressure ; b, tracing from needle passed through chest wall into ventricle ; c, thoracic movements ; d, dia
phragmatic respirations ; e, time in 10 seconds ; /, signal. The horizontal line at b is 10 mm. below the abscissa of blood
pressure.
the arterial pulsations seen near the end is a result of slowing of the cardiac rhythm).
This rise of pressure therefore must be due to arterial constriction caused by the drug.
The chloroform was given as concentrated vapour, producing abolition of corneal reflex
in one minute and death in about four minutes ; but how far the constriction was due to
direct action upon the arterioles, and how far to an action upon the vasomotor centre,
the experiment does not determine. The continuation of the rise in the tracing may
perhaps be ascribed to a condition produced by the commencing failure of respiration, the
vasomotor centre being stimulated by the venous blood; especially as it is accompanied
by a certain amount of cardiac inhibition. Such asphyxial rise may be very marked
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
321
when the respirations become shallow early in the administration, as is shown
strikingly in the tracing given in fig. 6.
While cases showing such a marked subterminal rise are uncommon, it is not
unusual to find a subterminal arrest of the fall of bloodpressure, so that the curve
remains for a minute or two at the same level, or shows a more gradual fall than
immediately before and immediately after. Such arrest of fall, when unaccompanied
by failure of respiration, may also be explained by the constricting action of the drug
on the arterioles, acting either through the vasomotor centre or directly. This con
Fig. 7. — Dog, weight 7000 g. Effect of inhalation through trachea tube of air nearly saturated with chloroform vapour.
The uppermost tracing (A) is that of the bloodpressure ; the second tracing (B) is costal respiration ; the third (C)
abdominal respiration ; the fourth, time in 10 seconds ; and the fifth, the signal marking when chloroform was admitted
and stopped.
In this experiment the heart failed before the respiration, and about 30 seconds later showed spontaneous recovery,
which was, however, only temporary. There was no recovery of respiration.
striction, although insufficient entirely to compensate for the continual and gradual
weakening and slowing of the heart which is going on the whole time, interferes
with the continuous and uniform fall of pressure, which would otherwise show itself.
At a much later stage the ventricular contractions, although greatly weakened,
produce large fluctuations of pressure in the arterial system, which is then com
paratively empty, owing to the accumulation of blood in the great veins and in the
dilated heart cavities.
322
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
Effect of Chloroform upon the Heart.
Our experiments abundantly illustrate the fact, which is now unreservedly conceded,
that of all ordinary anaesthetics chloroform is the one which produces the most
deleterious effect upon both heart and respirations. Nothing is more striking than the
comparison of tracings from chloroform experiments, such as those shown in figs. 5, 6,
and 7, with one in which ether is the anaesthetic agent (fig. 8). We have further
Fig. 8. — This tracing was taken from the same animal as rig. 7, and immediately before it. Air saturated or nearly saturated
with ether was inhaled through the trachea tube, between the two points marked by the signal. Previously to this tbe
dog was very lightly anaesthetised with chloroform, corneal reflex being present. Compare the effect of ether upon the
bloodpressure and respiration with chloroform, in these tracings.
investigated the action of chloroform upon the heart in situ with the chest opened, after
the method used by M'William ; * and also after removal of the organ from the body,
with the coronary vessels perfused by Langendorff's method. f
In confirmation of previous observers, we find that the effect upon the organ when
its nervous connections are severed, or when the activity of the vagus is abolished by
atropine (fig. 9), is to produce a gradual weakening of the contractions (without any
marked slowing, although this may appear towards the end of a fatal experiment)
Jov,r. Phys., vol. xiii. p. 860, 1892.
+ Pfluger's Arch , vol. lxi. p. 291,'1895.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
323
affecting both auricles and ventricles ; but the former are affected much more than the
latter. This is also the case when the cardioinhibitory centre is refiexly stimulated,
even in presence of atropine (fig. 10). On the other hand, when the nervous connections
are intact, the administration of a strong dose of chloroform not only greatly weakens
the force of the cardiac contractions, and thereby causes a marked and progressive fall of
bloodpressure, but often, after a primary acceleration, produces gradual slowing of the
rhythm, and this is frequently followed by abrupt and complete cardiac arrest (figs. 6
and 7). The abrupt cessation of contraction may affect both auricles and ventricles
simultaneously, or the auricles may first stop ; the ventricles, either at once or after a
: ■ 1 1 ; : 1 1 1 i i ' ! ' ; ! ' ' "; i :_;:i 1 1 i :j ! ■ r /■ " .'■!_' ; v : ■ li! !JJj;i t USi) iUMW ' W JJ aiU^O^JLU^JjlU^W JWU*UJUjJ>jijUaa.a>>^^JUU,' i J
iilililli'iiiiilN'r, '■[.;(,? :. ''■'■!:i!ili
Fig. 9. — Dog, weighing about 10 kilos. The animal had received some 3 hours previously  00054 gramme (^1q gr.) atropine
sulphate administered hypodermically. The effect of this was to abolish arrest of the heart on stimulation of the cardio
inhibitory centre (see fig. 10), whilst permitting a diminution in force of the beats, especially of the auricle. It will be
seen from these tracings that exactly the same effect is produced in an animal under the influence of atropine by chloro
form alone in strong dose as is caused by reflex excitation of the cardioinhibitory centre, except that the result is attained
more gradually.
a, auricular tracing ; b, ventricular tracing ; c, bloodpressure (femoral) ; d, respiratory movements of the thorax, which
are continued in spite of the fact that artificial respiration is carried on by perflation ; e, time in 10 seconds ; /, signal.
short period of arrest, resuming their action with a rhythm of their own (figs. 6, 7,
and 11).
The effect entirely resembles that produced by vagal excitation, with the exception
that vagal excitation does not, as a rule, by itself produce permanent arrest of cardiac
action. But at a certain stage of chloroform anaesthesia the arrest produced by
artificial excitation of the vagus may be permanent, or so prolonged as to lead to death.
The cessation of the heart's action brings the blood pressure to zero, and by arresting the
324
PROCESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
Eil
Fio. 10. — Testing of cardiac centre by stimulation of central end of
one vagus — the other being intact — in the dog from which
the tracing given in fig. 9 was taken, and immediately
previous to that tracing.
a, b, c, d, e, /as before. Notice, as with chloroform, the
great diminution in force of the auricular beats, but without
arrest or slowing, the diminution in force of the ventricular
beats being hardly perceptible. The fall in bloodpressure
may be in part due to depressor action, but a similar fall
was obtained by stimulation of the peripheral cut vagus.
Fig. 11. — Chloroform inhalation. Showing cessa
tion of auricle before ventricle, the latter then
assuming its independent rhythm, with larger
excursions.
a, auricle ; b, ventricle; c, bloodpressure
(the alignment of this pen is a little in ad
vance of the others); c 1 , zero of bloodpressnre ;
d, time in 10 seconds. Artificial respiration
by perflation.
Fig. 12. — Effect of excitation (with coil 100) of the peripheral vagus (second signal) during moderately deep chloroform
anaesthesia, strong chloroform vapour having just previously been administered during 1^ minutes (first signal). The
result is seen to be an immediate heart arrest, with the bloodpressure falling to zero ; the respirations cease 30 seconds after
the heart has stopped, but are only gradually arrested. In this case the heart (ventricle ?) begins to escape from the arrest
after 40 seconds, beating at first very slowly, but after a minute faster : as the heart recovers, the respirations are also
renewed.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
325
circulation causes also a failure of the respiratory centre. This cause of respiratory
failure has been clearly demonstrated by Embley.*
The complete similarity between the tracings obtained under the influence of
chloroform alone on the one hand (when a concentrated vapour is inhaled), and of
..^^(♦lilllWww..
Fig. 13. — Tracing showing the effect of moderate vagus stimulation (coil 100) during deeper chloroform anesthesia. In this case
the excitation was not applied until the respiratory movements had nearly ceased. The effect is to produce complete arrest
of the heart, which, however, shows a beginning of escape from the arrest after the lapse of about a minute, and 20 seconds
later resumes beating slowly and feebly, and with but little effect upon the blood pressure. Respirations are not resumed
spontaneously, but the animal was recovered 5 minutes after the respirations had ceased, by artificial respiration (compression
of thorax) continued during about 2 minutes. The first signal mark shows the period of strong chloroform administra
tion : the second that of vagus stimulation.
chloroform plus artificial vagus excitation on the other, shows conclusively that in the
former case, as in the latter, the actual cause of the arrest of the heart (and of the
Fig. 14. — Tracing showing the effect upon the heart (1) of weak and (2) of stronger vagus excitation during extreme
chloroform anaesthesia. The first signal mark shows the period of strong chloroform administration : the second that of
weak vagus stimulation : the third that of stronger vagus stimulation. The chloroform was given until the respiration
had ceased and the bloodpressure had fallen to 20 mm. Hg. Excitation of the vagus by induction shocks of very
moderate strength produced only a momentary arrest of the heart, but stronger excitation caused instant and permanent
arrest.
respiration as a secondary effect) is inhibition excited through the vagus. Thus
figs. 5, 6, 7, and 15 show such a cardiac and respiratory arrest produced by strong
chloroform alone, and figs. 12, 13, and 14 the same phenomenon produced under
varying degrees of chloroform anaesthesia by vagal stimulation. In figs. 12 and 13 it
is seen that the ventricle has escaped from the inhibition and has resumed contraction
* Op. cit.
326
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
with an independent (slower) rhythm. In all such cases as these recovery may be
effected if artificial respiration be started soon (fig. 15), before the respiratory centre
has been allowed to remain too long under the influence of the chloroform plus anaemia ;
or if the anaesthetisation be not very deep, there may be spontaneous recovery (fig. 12).
In other cases the ventricle does not escape spontaneously (fig. 14) ; but it may be
caused to contract by rhythmic compression through the chest wall (fig. 15). The
arrest of the heart (and, secondarily, of the respiration) is due therefore to an excitation
by chloroform of the cardioinhibitory centre.* Arloing showed that it does not occur
with cut vagi, and this is also emphasised by Embley ; moreover, it does not occur if
a small dose of atropine has been previously given (see below, p. 328), and it may also
fail to be apparent after prolonged ansesthetisation with chloroform in moderate dosage.
It fails to occur also in certain individuals, which seem to be less susceptible than
others to the cardioinhibitory effects of the drug. Instances are shown in fig. 22, A, and
— ,\1); .V lj . v " '■" • ■■ •■J .*.' ! ■' J>'' ' >.*.^ ^i«>^ HH L i wu^.iwA^HH •
wmtmtmm
Si;;;;;
Fig. 15. — Tracing showing (secondary) inhibition of heart from strong chloroform inhalation, with simultaneous cessation ot
respiration. Recovery, after 30 seconds' arrest, by means of artificial respiration effected by chest compression.
a, respiration ; b, arterial pressure ; c, time in 10 seconds ; d, signal line (2 mm. above abscissa of bloodpressure).
also in fig. 16; the latter from a dog in which the inhalation of air strongly charged with
chloroform vapour was pushed until respirations had ceased, the heart continuing to
beat with great regularity five minutes longer, but during the last three minutes at a
slower rate (probably the result of independent ventricular action).
In connection with this subject, we have investigated the effect of vagal excitation
upon the heart in different stages of chloroform anaesthesia, and the effect of small
doses of atropine upon the result of vagal excitation. In light anaesthesia, an
adequate stimulation of the vagus produces, as in absence of anaesthesia, complete
arrest of cardiac movements with a fall of bloodpressure to zero. But even if the
excitation be continued, this condition does not last, for although the auricles may
* The above tracings make it abundantly evident that the assertion of Lawrie {Lancet, 1890, vol. i. p. 1393),
founded on the report of the Hyderabad Commission, " that sudden death from stoppage of the heart is not a risk
of chloroform itself," is completely erroneous.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
327
remain quiescent, the ventricles escape from the inhibition and resume contracting, at first
very slowly, afterwards more quickly, although often somewhat irregularly, so that the
bloodpressure again rises. On ceasing to excite the vagus, the heart beats more rapidly
and more strongly, and there may be a temporary rise of bloodpressure above the
normal average just before the excitation. This characteristic effect is shown in fig. 18.
In chloroform anaesthesia the effect upon the heart of vagal excitation is more
pronounced and permanent. * The complete arrest of cardiac action may last long
enough to cause a concomitant arrest of respiration, and when this occurs, even if the
Fig. 16. — Shows a tracing of respiration and bloodpressure under inhalation of air strongly charged with chloroform vapour.
This tracing illustrates the type of result obtained when cardiac inhibition does not occur to the extent of causing
complete arrest of the heart, but merely a slowing (which may be suddenly increased), the heart failing quite gradually.
The chloroform inhalation lasted 4J minutes, at the end of which period respiration had ceased, and was not again
renewed until the heart had nearly stopped, when a ' staircase ' group of 25 slow gasping respirations showed them
selves — the socalled 'respirations of the deathagony.' The lid reflex, which was present immediately before the
inhalation began, disappeared within one minute.
ventricles resume action, their rhythm is very slow, and their force insufficient to
raise the bloodpressure much, so that, as a rule, respirations are not spontaneously
resumed although artificial respiration may effect recovery. Or it may happen,
especially with a strong excitation, that the recovery of the ventricles does not occur
at all, and even heart massage, which can be effected by compressing the chest, to
which the ventricle may at first respond, may be incapable of causing it permanently
to resume its action.
* Dastre states that this increase of vagal excitability under chloroform was first noted by Vulpian (0. r. Soc.
Biol, 1883, p. 243). According to Fean90ISFranck (ibid., p. 255), it disappears with increase of anaesthesia, but this
is not in accordance with our experience so far as concerns direct excitation.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 12.) 49
328 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
If the chloroform be pushed until respirations cease altogether and the blood
pressure is reduced to a few millimetres of mercury, the only sign of life being the
slow beat of the heart, adequate vagal excitation will still cause inhibition, which
under these circumstances lasts as long as the excitation ; the removal of excitation
being immediately followed by a resumption of the slow, weak beats. Such inhibition
can be obtained as long as there is any perceptible beat (fig. 17).
We have also restudied the effect and dosage of atropine in preventing inhibition
through the vagus. The results obtained are illustrated in the tracings reproduced in
fig. 19. (See also fig. 10 for its effect on reflex vagal excitation.)
If a dose of sulphate of atropine of '00002 gramme per kilo, be given hypodermi
cally in the dog, the effect upon the vagus is manifest about fifteen to twenty minutes
Fig. 17. — Effect of moderate vagal stimulation in the
last stage of chloroform anaesthesia, the respira
tion having long ceased, and the heart beating
slowly, feebly, and irregularly. The signal marks
the period of vagal stimulation. (The alignment
of the signal is a little too much to the left.)
It will be seen that excitation of the peripheral
vagus still causes arrest of cardiac action, which
is at this stage probably entirely ventricular.
after administration, and lasts about three hours. The result of such a dose is in
some cases to abolish for a time all vagal influence upon the heart (fig. 19, I.). But in
most cases, although there is not complete abolition, nevertheless the strongest vagal
excitation is unable to produce, in any stage of chloroform ansesthesia, complete cardiac
arrest (fig. 19, II. to VI.). There may be, even with comparatively weak excitation, a
slowing of the heart and a consequent fall of bloodpressure ; but it is no greater with
strong than with weak excitation, and is never accompanied by respiratory arrest, unless
in using a very strong excitation there is escape of current to the central end of the
nerve. This peculiar condition, in which vagal excitation is unable to cause arrest, but
only slowing and diminution in force of the heart, persists for nearly three hours, the
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
329
slowing becoming as time proceeds gradually more marked, and the consequent fall of
bloodpressure lower. But cardiac arrest does not show itself until the influence of the
atropine has completely passed, and the former conditions can be then restored by a
similar dose.
There is no reason to believe that the human subject is less susceptible to the
influence of atropine than the dog, and the opinion which has been expressed, that an
enormous dose would be required to abolish the power of the vagus to cause cardiac
arrest, appears therefore to be erroneous.*
Fig. 18. — Normal effect, iu the dog, of vagus stimulation of moderate strength (coil 100 mm.) with light anaesthesia, showing
the tendency of the heart to escape from the inhibition. This was taken immediately before and from the same dog as
the (reduced) tracings shown in figs. 12, 13, and 14, but chloroform was administered in the interval, and the tendency to
inhibition is seen in these to be much more pronounced.
Since abrupt arrest of the heart and of respirations can be absolutely prevented by
prior administration of a small dose of atropine, the conclusion forces itself upon'us
that the precaution of such administration is one that should never be omitted.
Atropine cannot, as we shall see, prevent death where a dose of chloroform
sufficient to produce paralysis of respiration and complete " paralytic dilatation " of
* See on this subject remarks by H. C. Crouch and T. G. Brodie in Trans. Soc. Ancesth., vol. vi. pp. 70 and 81,
1904. J. Harley (Brit. Med. Journ., vol. ii., 1868, p. 320) recommended a dose of from ^g grain to fa grain in man,
Dastre (Soc. Biol., 1883, p. 242) states that a dose of atropine amounting to  0015 gramme ( = fa grain) is sufficient
for the purpose indicated. Langi^ois and Maurange (Arch, de Phys., 1895, p. 692) recommend the employment^
oxysparteine in place of atropine.
III.
IV.
VI.
Fig. 19. — Tracings to illustrate the influence on cardiac inhibition by vagus excitation of a small dose of atropine sulphate
('00054 gramme = T £„ gr.) administered to a dog weighing 28 kilog. =61 J lbs.
I., tracing taken 15 minutes after the atropine was administered by injection into the pleural cavity ; II., 30 minutes
after ; III., 45 minutes ; IV., 1 hour ; V., 1^ hours ; VI., 2\ hours after administration.
The bloodpressure and extent of anssthetisation are approximately the same in all.
The strength of stimulus was the same in all, and was adequate to produce strong inhibition in the absence of atropine.
Note that this effect is abolished in 15 minutes, and does not reappear in the same form and extent during at least 1\
hours, although there is a gradually increasing amount of inhibition shown as the atropine is becoming eliminated. But
even 2£ hours after the injection the strongest stimulus (coil at 0) failed to produce any more effect than that shown in VI.
a, bloodpressure curve ; b, respirations ; c, time in 10 sees. ; d, signal showing period of vagal excitation.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 331
the heart has been administered, but this is a condition which no ansesthetist has any
right or occasion to produce. On the other hand, it can prevent the sudden cardiac
arrest which may show itself even at a comparatively early stage of chloroformisation,
and which may be produced by the drug itself acting on the cardioinhibitory centre,
with (or without) the assistance of an effect reflexly produced upon that centre by
the operation which is proceeding ; or by some other concomitant, such as the incidental
occurrence of dyspnoea, which is well known also to cause inhibition through the vagus.*
Whether the vagi be cut or their terminal branches blocked by atropine, or whether
they be left intact, the ultimate effect of chloroform upon the heart, if its inhalation be
pushed, is to produce complete arrest in diastole, a condition being observed which has
been termed " paralytic dilatation." The condition is a peculiar one, for the cardiac
muscle is not only paralysed and incapable of contracting spontaneously, but is in a
permanently refractory condition, and incapable of responding to stimuli of any sort.t
To all forms of direct stimulation the heart gives no response, although the
muscular tissue is not dead, and it suffices to remove the chloroform by producing a
flow of unpoisoned blood or circulating fluid through the coronary vessels to restore its
rhythmic contractility and its power of responding to artificial stimuli.^ This shows
that the refractory condition is due to the influence of the drug upon the heart, and
it is commonly assumed by writers upon the subject that chloroform enters into
combination with the contractile substance of the cardiac muscle and thereby deprives
it of irritability. That this assumption is not justified is clear from the fact that no
such effect — in doses which are more than sufficient to paralyse the heart — is produced
upon either skeletal or upon plain muscular tissue. It is impossible to believe that the
chemical constitution of these forms of contractile tissue is so different from that of
heart muscle, that the one combines with chloroform and is thereby rendered devoid
of irritability, whilst the others show no tendency to combine with or to be materially
affected by the drug. It is much more probable that the effect produced is one of
excitation of the terminations of the inhibitory nerves, the heart being thereby
rendered irresponsive to stimuli. The argument that may be urged against this
hypothesis, that if this were so the effect of chloroform in paralysing the heart would
be prevented by atropine, is met by the statement that, although atropine blocks the
* An instance of the lastnamed complication is illustrated in fig. 20. In this animal the breathing was laboured,
owing to obstruction of the airtubes by mucus. There was marked dyspnoea, and the heartbeats were very slow
and even arrested whenever the dyspnoea became intense. The violent respiratory efforts succeeded from time to time
in clearing the airpassages, and this was followed by partial recovery. This pronounced inhibition was due to
asphyxia, which, if more marked than in the instance given, may lead to entire arrest of the heart. Such inhibition
from asphyxia does not occur with cut vagi. The condition is one which is not unfamiliar to anaesthetists, who are
cognisant of its cause and danger. It is not liable to occur if a prior dose of atropine be administered, partly on
account of the effect of this on the vagi and also because atropine tends to prevent the secretion of the mucus which
causes the obstruction to respiration. This reason for the administration of atropine will apply equally to ether as
to chloroform anaesthesia.
t Sherrington and Sowton (op. cit.), in the isolated and perfused cat's heart arrested by chloroform, obtained
a renewal of the contractions on stimulation of accelerator nerves. But it is doubtful if this could be obtained with
■a strong dose of chloroform.
X Sherrington and Sowton.
332 PROFESSOR E. A. 8CHAFER AND DR H. J. SCHARLTEB ON THE
inhibitory path, there is no conclusive evidence that it acts upon the inhibitory end
apparatus in the muscular fibres. According to this view the chloroformheart —
provided that the dose is insufficient to kill the contractile tissues generally— is in a
condition of active inhibition rather than in one of passive paralysis. In support of
this, it may be stated that although, if the chest be opened immediately after death,
the heart may be completely irresponsive to all forms of stimuli, after a little while it
often happens that it begins to respond and even exhibits feeble spontaneous contraction,
Fig. 20. —Cardiac inhibition produced during chloroform inhalation by dyspnoea resulting from accumulation of mucus in
airpassages.
A, bloodpressure ; A 1 , line 1 centimetre below the zero of bloodpressure ; B, costal respiration : C, diaphragmatic
respiration. The dyspnceic condition is shown by the extreme rapidity of the respiratory movements at the left hand of
tracing. About the middle of the tracing the obstruction to the passage of air was removed, and with the disappearance
of the dyspnoea the heart resumed its normal rate of rhythm.
although the chloroform has not been washed away. The phenomenon may be explained
if we assume that the inhibitory endapparatus has died before the contractile substance
of the muscular fibres.
To sum up this part of the subject, the conclusions which it appears justifiable to
draw regarding the causation of death from the effect of chloroform upon the heart
are as follows : — ( 1 ) Death may be caused in the earliest stage of administration by the
action of the drug upon the cardioinhibitory centre, the stimulation being reflex.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 333
For convenience of description, this may be termed " primary " inhibition. It is prob
able that it occurs but rarely in man, and some altogether deny that it can produce
a fatal result ; it is, however, impossible to explain the wellauthenticated instances
of sudden heart failure at the very commencement of administration without assuming
that under certain conditions such primary inhibition, which, when it occurs in other
cases, is usually quite evanescent, is occasionally persistent and fatal. #
(2) At a somewhat later stage of administration death is liable to occur from
sudden heart failure due to inhibition produced by the action of the drug on the
cardioinhibitory centre, aided by its action on the neuromuscular endapparatus of
the heart itself, and also by increasing venosity of blood caused by failing respiration.
The liability to this form of inhibition, which may be termed " secondary " (as well
as to that mentioned under 1), can be removed by the prior exhibition of a moderate
dose of atropine, which, by diminishing or abolishing the effect upon the heart of
excitation of the cardioinhibitory centre by chloroform, deprives it of the power to
produce sudden cardiac arrest. This precautionary measure was long ago suggested, t
and all recent work on the subject emphasises the importance of its adoption.
That the prolonged administration of chloroform itself tends to diminish its
excitatory effect upon the cardioinhibitory centre in the medulla oblongata is probable
from the fact that a dosage of chloroform can be given with impunity at the later
stages of a long operation which would be highly dangerous if given at earlier stages.
The respiration in these " secondary " cases of inhibition may stop simultaneously
with, or shortly before, or immediately after the heart. We have frequently succeeded
in effecting resuscitation by artificial respiration in animals, in which both heart and
respiration had completely stopped at this stage of poisoning after even a minute or
two of cessation of heartbeats, and in two cases as long as three and five minutes
respectively after complete cessation ; but in other instances we have failed to obtain
recovery after three minutes or more of cessation.
The two cases just referred to are of exceptional interest. In the one the animal had been under ether
for about an hour (the anaesthetic being inhaled through a Yshaped trachea tube), when chloroform, at first
with considerable intermixture of air, was substituted for the ether. The effect was to produce a gradual
fall of bloodpressure from 100 mm. Hg. to about 40 mm., after which both it and the respiration, which was
much shallower than under ether, remained nearly constant. After five minutes the lateral airinlet was cut
off, and the dog received a much stronger dose of chloroform. The result of this was immediately apparent
in a further fall of bloodpressure, and a slowing and irregularity of the respirations, which ceased altogether
about two minutes later, although the heart continued to beat regularly and the bloodpressure was maintained
and even rose slightly. About one minute twenty seconds after cessation of respiration the heart suddenly
* This mode of producing inhibition has been especially emphasised by Arloing (These, Paris, 1879), who describes
the effect of chloroform in producing heart failure in terms very similar to those which we have employed.
t Pitha (1861, quoted by Dastre) ; J. Harley, Brit. Med. Joum., vol. i., 1868, p. 320 ; Schafer, Brit. Med.
, Joum., vol. ii., 1880, p. 620. Fraser (Brit. Med. Joum., vol. ii., 1880, p. 715), BrownSequard (G. r. Soc. Biol., 1883,
p. 289), and Dastre and Morat (Lyon Med., 1882, and C. r. Soc. Biol., 1883, pp. 242 and 259) have made a similar
recommendation, but have suggested the addition of morphia, and this combination has often been used (first
systematically by Aubert, C. r. Soc. Biol., 1883, p. 626). But morphia is in some ways antagonistic to atropine, and
tends by itself to exalt the irritability of the cardioinhibitory centre. Without atropine it would undoubtedly
increase the danger of heartarrest in chloroform administration.
334 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
stopped and the bloodpressure fell to zero. Artificial respiration (by the pump) was now started, and main
tained for five minutes, and two injections of 4 c.c. adrenalin chloride (1 per mille) were meanwhile made
into the pleural cavity without the least result being apparent, the heart remaining quiescent and the blood
pressure at zero the whole time. Intermittent compression of the thorax was now substituted for perflation,
and the heart responded to this by a pulsation with each compression, although in the intervals the blood
pressure returned to zero. After continuing the compression for a little over a minute, the heart commenced
beating spontaneously and the average bloodpressure rose to 50 mm. Hg. It required, however, another
three minutes of artificial respiration before the diaphragm began to act and the intermittent compression
could be desisted from for a time ; but even then (although no more chloroform had been inhaled) the
respirations again gradually failed and ceased after ten minutes. A second spell of intermittent compression,
lasting two minutes, now, however, effected complete restoration. When this was established and the lid
reflex had become brisk, the bloodpressure being 110 mm. Hg., chloroform was again administered in
strong form. The bloodpressure gradually fell. In two minutes the lid reflex had disappeared, and in
another minute respirations had ceased, followed in twenty seconds by complete cessation of heart
beat. Five minutes was now allowed to elapse, during which the animal was to all appearance dead.
Artificial respiration by chest compression was then recommenced, and two more doses of 3 c.c. adrenalin
chloride solution were successively injected. Five minutes after the artificial respiration was commenced
and immediately after the final dose of adrenalin, the heart began to beat spontaneously, and the blood
pressure, at first very low, gradually rose in about four or five minutes, during which artificial respiration
was maintained, to about 100 mm. Hg. Natural respiration was, however, not again resumed, the medulla
oblongata having to all appearance been deprived for too long a time in this instance of blood.
In the second dog a lethal dose of chloroform vapour was administered twice. The first time both heart
and respiration (the latter ten seconds before the heart) had stopped after three and a half minutes' adminis
tration. Half a minute later the chloroform was removed, and 4 c.c. of 1 per 1000 adrenalin chloride solution
was injected into the pleural cavity. This produced no apparent effect. Three minutes after cessation of
heart and respiration, chest compression was begun. Each compression produced a heart response, and the
bloodpressure rose from zero to a few millimetres. After four minutes' chest compression another similar
dose of adrenalin was injected into the pleural cavity, chest compression being continued. The blood
pressure then began to recover, the heart now beating slightly more rapidly than the chest compression, but
natural respiration (diaphragm) was not resumed until another six minutes had elapsed. As in the last
case, however, the natural respirations gradually became shallower and slower again, although no more
chloroform was given, and fifteen minutes later ceased altogether, the heart and bloodpressure also becoming
weak and low ; the administration at this stage of a decoction of pituitary, and later of another dose of
adrenalin, with the idea of restoring the heart's action, produced no visible effect. After two minutes'
cessation of respiration (the heart still beating feebly), recourse was again had to chest compression and then
to artificial respiration by the pump. This was very soon followed by recovery of heart and bloodpressure,
and a few minutes later natural respirations were resumed and maintained, and artificial respiration was
discontinued ; recovery was, in fact, complete.
(3) In late stages of administration the heart is paralysed by the direct effect of the
drug, acting either upon its muscular tissue (as is usually assumed), or (as we believe)
by exciting the neuromuscular inhibitory endapparatus, and through this rendering
the muscular tissue nonexcitable. This effect can probably only occur with a consider
able dosage of chloroform in the blood, and the respiratory centre is invariably first
paralysed, so that the respirations become slow and shallow and cease before the heart ;
the time difference between the cessation of heart and respiration being considerably
longer than when the cessation occurs early in the administration. This final effect
upon the heart is not antagonised by atropine. The heart is found to be entirely
inexcitable, and no treatment is of any avail short of removal of the poisoned blood
from the coronary vessels and the substitution of blood free from chloroform. It is
conceivable that this substitution might be done by heart massage, or even by com
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 335
pression of the thorax in attempting artificial respiration ; and, as a matter of fact,
Schiff, Batelli, and others * have succeeded in restoring the circulation by heart
massage combined with artificial respiration (both in dogs and cats, and in one or two
rare cases in the human subject), even although a considerable time (fifteen to twenty
minutes) had elapsed after complete cessation of the circulation. We ourselves have
not succeeded in restoring an animal after the condition here described appeared to be
fully established, and we should be disposed to regard the possibility of resuscitation
in such cases as remote.t
Antagonising Agents.
(1) Atropine. — It is apparent from the results obtained by other experimenters, as
well as from our own observations, that the chief danger to be guarded against in the
administration of chloroform is the inhibitory influence which it produces upon the
heart. As we have already pointed out (pp. 328, 329), this influence can be in great
measure controlled by the prior administration of a moderate dose of atropine, at least
in so far as the primary and secondary instances of inhibition are concerned, and these
are the most dangerous because they are apt to occur without the warning which
manifests itself in the case of the final heart paralysis, by the prior arrest of respiration.
Atropine is therefore to be placed first in the list of antagonising agents ; a dose of
y^q gr. to 3^ gr. for an average man being administered hypodermically half an hour
before the administration of chloroform.
(2) Adrenalin Chloride. — The employment of this has been suggested in chloroform
poisoning by Gottlieb.;}; In the two instances which we have recorded on pp. 333 and
334, which were attended by an entirely unusual measure of success so far as resuscitation
after apparent death from chloroform had occurred, we administered successive doses
of adrenalin, injected into the pleural cavity, as part of the treatment. These happened
to be the first two experiments of the series undertaken by us, and we were led to ascribe
much of the success which attended them to the use of this drug, and formed high hopes
of the value of its administration in cases of chloroform poisoning. Subsequent experi
ence showed, however, that adrenalin by itself is of little or no avail to restore a heart
paralysed by chloroform, and even in conjunction with other remedial measures — of
which the most important is without doubt artificial respiration by chest compression —
we are not in a position, as the result of a number of trials, to affirm that it is able
materially to contribute to the process of resuscitation.
(3) Ammonia Vapour. — Einger § first showed that in the frog's heart ammonia acts
as a direct antagonist to chloroform, and may even set in activity a heart which has
* For references see M. Bourcart, Rev. m,4d. de la Suisse Romande, October 20, 1903.
t This is no doubt the condition referred toby Richet (Diet, de physiol., article " Anesthesie," 1895, p. 523)
when he avers that when cardiac syncope occurs artificial respiration never succeeds in effecting restoration ; for
the statement does not apply to the syncope caused by the secondary inhibition previously referred to.
X Arch.f. Path. u. Pharm., Bd. 37, p. 98, 1896. See also Bibdl, Wien. Hin. Wochenschr., 1896.
§ Practitioner, vol. xxvi. p. 436, 1881.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 12). 50
336
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLTEB ON THE
been arrested by chloroform ; and since it also has a stimulating action on the heart and
respiratory centre, it is likely that it may prove useful as a restorative in cases in which
the pulse and breathing have not altogether ceased. We have investigated the effect
in dogs of causing them to inhale a mixture of chloroform vapour and ammonia, made
either by dropping chloroform and ammonia upon the cottonwool of the inhaling
bottle, or by mixing chloroform in definite proportions with alcoholic ammonia, using
for this purpose a solution of ammonia in absolute alcohol containing 6 "8 per cent, of
Fi«. 21.— Instantaneous heart failure caused by inhalation, at the moment marked by the signal, of air charged with vapour
from a mixture of 20 c.c. chloroform and 5 c.c. ammoniated alcohol.
A, blood pressure ; B, respirations. The latter continued to show themselves at a slow rate for 3 minutes after the
heart had stopped.
ammonia, prepared for us by Messrs Duncan & Flockhart. A mixture of chloroform
and ammonia vapours, even if it contain a comparatively small proportion of ammonia,
is too pungent to be administered in the first instance, the irritation it causes to the
sensory nerves of the mucous surfaces rendering it practically irrespirable. And if the
proportion of ammonia be considerable, this excitation may result in powerful cardiac
inhibition, and the heart may instantly and permanently stop (fig. 21). If, however,
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
337
the proportion of ammonia be less and the animal be already completely under the influ
ence of an anaesthetic, the effect of the addition of the vapour of ammonia or ammoniated
alcohol to the chloroform inhaled is strikingly beneficial. The bloodpressure falls but
slightly even during a prolonged period of inhalation — the strength of the heart is well
maintained — and the tendency to failure of respiration, which is so marked a feature
}}MVMM!MUMMUMMMBMUMMMiMMm^mmmm!m!i,
WWIWN^^
B
UllMlillilUUIlUi
1,1 111 ImiiiiiiitiiiiUiii
■ fAiAf.iiiii.ifiW.ii 'iiViV.ii.ii.
Fig. 22. — Tracings showing in a comparative manner in the same animal the difference of effect between inhalation (A) of
pure chloroform, (B) of a mixture of chloroform with ammoniated alcohol (9 to 1 ), and (C) of a mixture of chloroform and
absolute alcohol (9 to 1).
Note in A the rapid fall of bloodpressure and the speedy failure of respiration ; in this case the heart continued to
beat after the respirations had ceased. After a respiratory arrest of more than a minute, during which the heart showed
strong tendency to inhibition, artificial respiration by chest compression was resorted to (the beginning of this is shown) :
in rather more than a minute the bloodpressure rose — the natural respirations were then resumed. Note in B and C the
very slow and slight fall of bloodpressure, and the complete maintenance of respiration during the whole time of adminis
tration. In all three cases the air of respiration was charged as strongly as possible, at the ordinary temperature of the
laboratory, with the vapour to be inhaled.
a, respirations ; b, bloodpressure ; c, time in 10 sees. ; d, signal showing period of administration. In all cases there
was distinct corneal reflex immediately prior to the administration, and this disappeared within 1 minute.
338
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
of strong chloroform inhalation, is considerably diminished. The comparative effects
of inhalation of chloroform alone and of chloroform plus ammoniated alcohol are shown
in the tracings A and B reproduced in fig. 22, and the beneficial results of substituting
a mixture of chloroform containing alcoholic ammonia for pure chloroform are illustrated
in the tracing given in fie;. 23.*
(4) Alcohol Vapour. — In order to determine how much of the beneficial effect of
the mixture of alcoholic ammonia with chloroform was due to the alcohol used as a
Fig. 23. — Beneficial effect upon bloodpressure, heart, and respiration of substituting ammoniated alcohol and chloroform (1 to
9) for the pure chloroform which was being administered to a dog.
a, bloodpressure ; b, heartbeats, recorded by a needle passed through chest wall ; c, respiration ; d, time in 10 sees.
Notice the increasing strength of the heartbeats and of the respiratory movements.
It is also seen that the dropped heartbeats due to vagal inhibition which were occurring during chloroform alone
gradually disappear as the result of adding ammoniated alcohol to the chloroform.
vehicle for the ammonia, we next proceeded to investigate the results of using for
inhalation the vapour given off from mixtures of chloroform and alcohol. We were
somewhat surprised to find that the results were nearly as beneficial when alcohol alone
was used, as when alcoholic ammonia vapour was employed. The difference between
the effect produced upon bloodpressure and respiration by inhalation of pure chloroform
in the one case, and by inhalation of a mixture of chloroform containing 1 part in 10
* The addition of ammonia gas to the chloroform to be used for inhalation was advocated by J. Duncan
Menzies {Brit. Med. Jour., vol. ii., 1895, p. 871).
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES.
339
by volume of absolute alcohol, is exemplified in fig. 22 (A and C). It will be observed
that the effect of the addition of 1 part absolute alcohol to 9 parts chloroform is to largely
prevent the fall of bloodpressure, which is recognised as being one of the most serious
dangers attendant on chloroform inhalation, and at the same time to maintain the
respirations at a force and frequency very little less than normal. The administration
was made in both cases on the same animal by the same method, and using the same
quantities of the solutions.
It may also be observed how much more readily recovery takes place after removal
of the mixed vapour than after removal of the chloroform. On the other hand,
disappearance of the lid reflex occurs a little sooner when pure chloroform is used, but
the difference is not great.
We have tried other proportions of alcohol and chloroform, but have obtained no
i!te %i( te
*^mm
mm ^m mik
liKSS
Bite
Fig. 24. — Administration by inhalation of air strongly charged with pure chloroform during 3 minutes to dog weighing 11
kilog. which had received (1 J hr. and If hr. previously) two doses of '00027 g. (t^ gr. in all) atropine sulphate. The vagus,
tested immediately before this tracing was taken, gave no slowing and only a slight fall of pressure, even with coil at 0.
Notice (1) the preliminary rise of bloodpressure due to excitation of vasomotor centre, succeeded by (2) a rapid and
regular fall, less steep towards the end ; (3) entire absence of slowing of pulse ; (4) increase of rapidity, but decrease of
excursion of respiratory movements, which became irregular, and eventually hardly perceptible.
a, bloodpressure curve ; b, respirations ; c, time in 10 sees. ; d, signal of chloroformisation and abscissa of blood
pressure. The heart was still beating 4 minutes later, but the bloodpressure was at zero, and the respirations had wholly
ceased. The animal was then subjected to artificial respiration by chest compression, and in 1 m. 40 sees, natural
respirations were resumed, and the heart and blood pressure rapidly recovered. The tracing shown in the next figure
was taken prior to this one.
better results. Indeed, with a 20 per cent, alcohol chloroform the respirations appeared
to be more affected than with the 10 per cent, mixture.
The beneficial effect can hardly be due to the mere dilution of the chloroform
vapour by alcohol vapour ; moreover, dilution with ether has not this effect, but
the result is then practically the same as is obtained with undiluted chloroform.
It is therefore to be ascribed directly to the beneficial action of the alcohol on the heart
and respiratory centre. We are of opinion that a mixture containing one part by
volume of absolute alcohol to nine parts of chloroform should be used when chloroform
is indicated as the anaesthetic, since these results show that it is far safer in its action
than pure chloroform. There seems reason to believe that the greater safety of the
A.C.E. mixture over chloroform depends upon the alcohol it contains, and that the ether
is unnecessary ; it may further be noted that the alcohol in this mixture is in needlessly
340
PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE
large proportion.* The beneficial effect of adding absolute alcohol to the chloroform
used for inhalation is seen in atropinised as well as in normal animals. The respective
effects of administering pure chloroform, on the one hand, and chloroform containing 10
parts per cent, of absolute alcohol, on the other, to a dog weighing 24 lbs., which had
previously received two successive doses of '00027 gramme of atropine sulphate, are
illustrated by the tracings shown in figs. 24 and 25.
A mixture which is frequently used by anaesthetists in place of the A.C.E. is one
of ether and chloroform without alcohol ; but from the facts here put forward it would
seem better rather to omit the ether than the alcohol. This remark is not to be
understood as implying that ether by itself is not a safe anaesthetic — far safer than
chloroform, however diluted — but merely that it has not the same antagonising influence
as alcohol upon the dangerous tendencies of chloroform.
Fig. 25. — Tracing showing the effect of the inhalation of air strongly charged with the vapour from a mixture of 9 parts pure
chloroform and 1 part absolute alcohol. The inhalation was continued during nearly 5 minutes. The tracing was
begun 10 minutes before that shown in the preceding figure, and is from the same animal (under the influence of
atropine sulphate). Notice, as compared with fig. 24, (1) the much more gradual fall of bloodpressure, which even after
nearly 5 minutes of administration still keeps fairly high, (2) the effect on the respirations, which are far less influenced
than by the pure chloroform, being well maintained during the whole time. On desisting from the inhalation, recovery of
bloodpressure was rapid, and the lid reflex, which had disappeared early during the inhalation, was brisk 5 minutes after
the chloroform and alcohol mixture had been removed.
a, bloodpressure ; b, respiration ; c, time in 10 sees. ; d, signal.
POSTMORTEM CONDITIONS AFTER DEATH FROM CHLOROFORM.
Although these conditions have been often described, it may not be out of place to
add our own experience and observations.
Heart. — In all the cases which we examined immediately after death, all the
cavities — with, sometimes, an exception for the left auricle — were distended with
blood, the right auricle and great veins of the thorax enormously so. The left
ventricle always contained a considerable quantity of blood, but rather less than the
right ventricle. If, however, the examination were made some little time after death,
the left ventricle was always found empty and firmly contracted. This change from
the full flaccid condition to the empty firm condition took place in one case within
twenty minutes, while in others it did not show itself for fortyfive minutes.
* Cf. on this subject, Quinquard, C. r. Soc. Biol., 1883, p. 425 ; and Dubois, ibid., p. 441.
ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 341
Lungs. — The pulmonary arteries are greatly distended with blood. Otherwise the
lungs usually present a perfectly normal appearance externally. But in cutting them
open we found, in six out of twenty cases examined, a considerable amount of frothy
mucus in the bronchial tubes.
Abdominal Viscera. — These exhibited marked venous congestion, especially well
seen in the liver, which may be greatly swollen and project beyond the thoracic cage*
It is thus exposed to some risk of rupture if artificial respiration be attempted by the
Howard method. This happened in one of our cases, although we were aware of the
danger, and always endeavoured to avoid it by compressing the chest well above the
liver.
All these appearances are very similar to those which result from asphyxia due to
deprivation of air, whether caused by drowning or otherwise. But they are produced
independently of any asphyxia caused by paralysis of the respiration by the drug, for they
show themselves equally when artificial respiration has been maintained by perflation,
and the drug has produced death solely by its action on the heart. Nevertheless, the
ultimate effect upon the heart of chloroform and of deprivation of air respectively is
strikingly similar. In both cases the final result is a condition of " paralytic dilatation,"
or, as we have preferred to term it, " inhibition paralysis," in which the heart is absolutely
refractory to all kinds of stimuli. In the case of chloroform the exciting cause is
doubtless the drug itself ; in the case of asphyxia, it is probably the carbon dioxide
which has accumulated in the blood and tissues, t
* To observe this condition of the liver and abdominal organs, it is necessary to open the abdomen before the
thorax. For if the contents of the latter be first laid bare, and any of the great veins injured, the congestion of the
abdominal viscera at once subsides, owing to the escape of blood from their vessels.
t Cf. on this subject the Report of the Committee on Suspended Animation, Trans. Med. Chir. Soc, 1904,
Suppl., p. 63.
( 343 )
XIII. — Continuants resolvable into Linear Factors. By Thomas Muir, LL.D.
(MS. received August 22, 1904. Read November 7, 1904. Issued separately January 13, 1905.)
(1) It is known that a continuant whose three diagonal are formed of certain
equidifferent progressions is resolvable into linear factors, the earliest specimens
placed on record being those of Sylvester and Painvin.* The object of the present
paper is to show that there are continuants of quite a different type which are also
so resolvable, and to expound a general mode of investigating the subject.
(2) The continuant of the n tk order ivhose main diagonal is
a, a + 2Vc, « + 22 2 c, a + 23 2 c,
and whose minor diagonals are
2(nl)b, («2)(6 + c), (n3)(6 + 2e),
n(bc), (n + !)(& 2c), (» + 2)(&3c),
is equal to the product of the n factors
{a + 2(nl)b} ■ {a + 2(?*3)6 + 2(2w3)c}
• {a + 2(nB)6 + 4(3n5)e}
• {o + 2(n7)6 + 6(2«7)c}
■ {a2(nl)& + (2n2)lc}
Taking for the purposes of illustrative proof the case where n = 5, viz.
a 246
5(6 c) a + 2c 3(6 + c)
6(6 2c) a + 8c 2(6 + 2c)
7(6 3c) a+18c 1(6 + 3c)
8(6 4c) a + 32c
and performing the operation
col x + col 2 + col 3 + col 4 + col 5
we find we can remove the factor a + Sb and write the cofactor in the form
a86 + 2c 3(6 + c)
26 12c a + 8c 2(6 + 2c)
86 7(6 3c) a + 18c 6 + 3c
86 . 8(6 4c) a + 32c
Performing now on this cofactor the operation
colj + 4 col 2 + 9 col 3 + 16 col 4
* [Sylvester, J. J.] " Theorerne sur les determinants de M. Sylvester," Nouv. Annates de Math., xiii. p. 305.
Painvin, . " Sur un certain systeme d'equations lineaires," Journ. de Liouville, 2 e ser., iii. pp. 4146.
Muir, Thomas. " Factorizable Continuants," Trans. S. Afr. Philos. Soc, xiv. pp. 2933.
(I)
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 13).
51
344
DR THOMAS MUIR ON
we obtain
(a + 46+1 in) 1 36 + 3c
4 a + 8c 26 + 4c
9 lb 21c a+l&c + 3c
16 . 86 32c a + 32c
of which the determinant factor is reducible to the threeline form
a 126 4c 26 + 4c
206 48c a+18c 6 + 3c
486 48c 86 32c a + 32c
The next operation
gives in like manner
colj + 6 col 2 + 20 col 3
(a + 20c)
1 26+ 4c
6 a+18c 6 + 3c
20 86 32c a + 32<
or
(a + 20c) : a 126 6c 6 + 3c
326 112c a + 32c
and finally the operation
colj + 8 col 2
enables us to change this twoline determinant into
(a 46+ 18c) 1 6 + 3c
8 a + 32c
or
(a46 + 18c)(a86 + 8c)
The desired result thus is
(a + 86)(a + 46 + Uc)(a + 20c)(a  46 + 18c)(a  86 + 8c)
(3) The continuant of the n th order whose main diagonal is
a, a + 2(l3)c, a + 2(24)c, a+2(35)c, . . .
and whose minor diagonals are
{nl)b, («2)(6 + c), (»3)(6+2c), . . . .
(n+2)(63c), (n + 3)(bic), (ra + 4)(65c), . . .
is equal to the product of the n factors
{a + 2(«l)6},
{a + 2(w3)6 + 2(2nl)c},
{a + 2('/«5)6 + 4(2ra3)c} ,
{a2(nl)6 + 6(«l)c} . (II)
This is established by proceeding in the same way as in § 2, the sets of column 
multipliers now being
1, 2, 3, 4, 5
1 , 4, 10, 20
1, 6, 21
1, 8
1
instead of
1, 1, 1, 1, 1
1, 4, 9, 16
1 , 6, 20
1, 8
1
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
345
and the resulting factors greater by
respectively.
0, 4c, 8c, 12c, 16c,
(4) Changing a into a + 2c, n into n— I, b into b + c in § 3 we have
a+ 2c (n2)(b + c) . . . . . .
(n+Y)(b2c) a + 8c (n8)(6 + 2c) . . . . .
(» + 2)(63c) a+18c (?i4)(6 + 3c) . . . .
(» + 3)(64c) a + 32c . . . .
= {a + 2(n  2)6 + 2(»  l)c} ■ {a + 2(n  i)b + 6(»  2)c}
• {a + 2(»6)6+10(»3)c}
But the continuant here is the complementary minor of the element in the place ( 1 , 1 )
of the continuant in § 2. Consequently by division we obtain
zMtlW (n2)(n + l)(b2c)(b + c)
a+2l 2 c
ffl + 22 2 c
\ h(2n2)(bnr. + c)( b + nc 2c)
a + 2(nl) 2 c
{a + 2(«l)6}{a+2(n3)6 + 2(2n3)e}{a + 2(n5)6 + 4(2n5)e} . . . ■ i\l\\
{a + 2(n2)6 + 2(»l)c}{o + 2(n4)6 + 6(n2)c} . . . .
(5) If in the results of §§ 2, 3 we annex f as a factor' to every term on both sides
that is independent of a, the identity is not interfered with. . . . (IV)
For, taking (in the fourth order, for shortness' sake) the continuant dealt with in
§ 2, and putting ct/f for a we have
3b
6(6  3c) ± + 6c 2(6 + c)
7(6 4c) il+16c b + 2c
8(6 5c) 4 + 30c
= (± + 66 V± + 26 + UcY^  26 + 20cY^  66 + 18c) ,
whence on multiplying both sides by/ 4 there results
a 36/
6(6 3c)/ a + 6c/ 2(6 + c)/
7(6 4c)/ a + 16c/ (6 + 2c)/
8(6 5c)/ a + 30c/
= (a + Qbf)(a + 26/+ 14c/)(a  26/+ 20cf)(a  66/+ 18c/) ,
as asserted.
346
DR THOMAS MUIR ON
(6) The sum of the elements of the main diagonal of either of the continuants in
§§ 2, 3 is equal to the sum of the factors into which the continuant is resolved. . (V)
This is true of any continuant of the form
a + x
Pi
</! a + x 2 p 2
q 2 a + x 3
that is resolvable into factors linear in a. By way of proof we have only to note (1)
that since the diagonal term is the only term of the continuant that contains either the
n th or (n— l) th powers of a, it follows that the coefficient of a n ~ x in the continuant
is x 1 +x 2 + x s + . . . , or 2a? say : and (2) that if a + ny , a + M 2 , a + M 3 , • • • be the
factors into which the continuant is resolved, the coefficient of a" 1 in their product is
/«! + m 2 + m 3 + • • • , or 2m say. We thus have
2.r = 2yu,,
and . *. na + 2.c = na + 2/x ,
as was to be proved.
(7) The full table of multipliers used in § 2 is found to be
1,1,1,1, l, ,1
1, 4, 9, 16, ,LC ril
1, 6, 20, ,yC r+li3
1,
•1 C
r+3, 7
— in other words, each multiplier is of the form
r n
— ^r+s1 ) 2sl ,
and the question next arises whether the continuant resolved in § 2 is the only one
which this set of multipliers is capable of dealing with. In order to make suitable
answer we have to ascertain the relations which must exist between the twelve
quantities
in the continuant
&,
P'2 > Pi ) Hi
P,
q, r.s
7i.
72 ' 73 . 74
a
24ft
5 7l
a + p
3/J,
6 7'2
a + q 2(3 3
7y 3 a+r ft
8y 4 a + s
in order that it may be resolvable into linear factors by means of the operations of § 2.
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 347
The performance of the first operation on the columns evolves the equations
n + 8f3 l = 5y, + a +p + 3/3 2 ,
= Qy 2 + a + q+2f3 3 ,
= 7y 3 + a + r + (3 i ,
= 8y 4 + a + s .
Then the factor a + 88 x being removed, the cofactor becomes expressible as a
determinant of the next lower order, viz.
a + p8/3, 3/? 2
6 yi 8^ a + q 2/3 3 .
8/3, 7y 3 a + r /3 4
 8/3, . 8y 4 a + s
and as by hypothesis the performance of the orjeration
colj + 4 col 2 + 9 col 3 +16 col 4
enables us to remove the factor a + p — 88 t + 12/3 2 we obtain three other conditional
equations, to be followed at the next stage by two others, and last of all by one. Of
these 4 + 3 + 2 + 1 equations the four first obtained are obviously best used to determine
p, q, r, s in terms of the B's and y's, the results being
p = 8/3 x  3/3 2  5 Yl ,
q = 8/3,  2/3 3  6y 2 ,
r = 8/3,  ft  7y 3 ,
a = 8/8, 8y 4 .
The remaining six form a very interesting set : after simplification they are
12ft  1 8/3, + 5/3 3 = 10 yi + 9y 2
64/3,  81/3 2 + l(3 i =  45 Tl + 35y 3
15/3,18^ = 10y, +7y 4
45/3 2  60/3 3 + 1 4/3 4 =  36y 2 + 35y 3
24/8 a 25/8, = 15y 2 +14y 4
P% " Pi =  Y3 + Y4 •
Taking the first three and using with them the multipliers 7, — 1, 1 respectively, we
find, on adding, that
5(& + Yi)  9(/3 2 + y 2 ) + 5(/3 3 + y 3 )  ( 4 + /3y 4 ) = 0;
similarly from the subset of two there results by subtraction
3(/3 2 + y 2 )  5(/3 3 + y 3 ) + 2(/3 4 + y 4 ) = 0;
and the final set, of course, is
By means, therefore, of these three derived equations we arrive at the proposition that
in the determinant under discussion the sum of any 8 and the corresponding y is
constant.
This being equivalent to only three equations, and other three being still un
accounted for, we put
Yl > 72 . Jz . Ji = °"  Pi > & ~ Pi > °" ~ As » °" _ Pi <
348 DR THOMAS MUIR ON
and learn (1) that one of the equations is not independent of the others, (2) that the
/8's are connected by the equation
i e i 2i8 2 + ^ s = 7(&2/3 3 + /? 4 ),
and (3) that a is expressible in terms of any three of the ,8's, for example,
o =  2/J, + 9/3 2  5/J s .
The conclusion thus is that in the continuant with which we started we can retain
any three of the /8's , and express in terms of these the fourth /8 , all the y's , and p , q ,
r , s , — thus obtaining a function of four variables which is resolvable into linear
factors.
(8) Had the determinant operated upon been of the sixth order, we should still
have found 0 =  2/^ + 9,8 2  5/8 3 and the first four /8's connected by the same equation
as in the preceding case, but there would have been a fresh equation of condition
connecting the second set of four consecutive /8's, viz.
/3 2 2/} 3 + &= 3(/S 3 2/3 4 + /3 5 ).
Similarly the case of the seventh order would be found to differ from that of the sixth
merely in having the additional equation
5(/3 3 2 i Q 4 + /3 5 )= Il(j8 4 2ft + j8 a )j
and so on.
As the result of all this we therefore affirm that — If the continuant
a 2(?i  l)/5j
ny 1 a + ]> (n  2)/3 2
(w+l)y 2 a + q (?i3)/3 3
(?i + 2)y 3 a + r
be resolvable into linear factors by means of the set of multipliers
llill
1 4 9 16
1 6 20
1 8
1
then (1) every four consecutive /3's are connected by a linear relation, viz.
l.(ft2/3 2 + /3 3 ) = 7(/J 2 2/3 8 + &),
3.(/? 2 2& + &) = 9(/3 8 2/3 4 + /5 6 ),
5( i 8 3 2/8 4 + /3 5 )= ll.(/3 4 2/3 5 + /3 6 ),
i/iws making all the fi's expressible in terms of any three; (2) all the y's are expressible
in terms of the same three /8's because of the fact that fi m + y m = — 2/3 l + 9/8 2  5@ 3 for
all values ofm; and (3) p, q, r, . . . are also so expressible because the sum of the
elements of any row of the continuant is constant . .... (VI)
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
349
(9) Instead, however, of taking a and three of the /3's as variables it is better to
take a and ^ , /3 2 — /3 2 , B x — 2/3 2 + /3 3 . Doing this and calling the last three quantities
b, c, d — a change implying the substitution
ft = b,
/3 2 = bc,
f$ 3 = b  2c + d ,
— we can by using the equations of condition obtain the requisite expressions for
^ , At , /3 5 , . • • , 7i , 7 2 j • ■ • • in terms of b, c, d. The theorem to which this course
ultimately leads is — The continuant
A a 2(»l))8 1 .
"7! A 2 (ra  2)/? 2 .
. (n + l)y 2 A, (nS)ft
(» + 2)y 3
A,
is resolvable into linear factors if
ft m = b  (m  \)r + x ■ od
y,„ = b + mc 
•2(2m 1)
(m+ \)m E ,
2(2ml)
A,„ = a2(ml) 2 c + — — ^_
• {2?»(m2) + 3ra}5tf ,
(VII)
(2m3)(2ml)'
^e s th factor being
a + 2(u2s + l)b  2(«l)(2n2a+l)e + (n  s + 2)(s  l)5d
For example, when n = 4 we have
a 23/;
i(b + a5d) a2c + 20d 2(bc)
5(6 + 2c 5^) a8c + 24d b2c + d
6(b + 3cM) al8c + 36d
= (a + 66)(a + 2b  10c + 20d)(a 261 2c + 3CW)(a  66  6c + 30d) .
On putting d = the /3's and 7's form equidifferent progressions, and the theorem
degenerates into that of § 2.
(10) Out of this effort to obtain greater generality an unexpected and curious
result arises ; for, whereas at first sight both members of the identity are functions
of the four variables a , b , c , d , it is found on careful examination that the righthand
side is expressible as a function of one less. In fact it is readily verified that the s th
factor given above can also be written in the form
{a + 2(nl)b}  2(sl){263c}  (ns + 2)(sl){ic5d}
so that the factorial expression for the continuant of the n th order, besides being
{a + 2(nl)b\ ■ {a + 2(n3)b2(2n3)c + ln5d}
■ {a + 2(n  5)6  4(2?;  5)c + 2(ra  l)5d}
■ {a 4 2(»  7)6  6(2»  7)c + 3(w  2)5d}
{a  2(?i  1)6  (2n  2) 1 c + (»  \)2 5d},
350
DR THOMAS MUIR ON
is also
X{X 2YlnZ}
• {X4Y2(w 1)Z}
• {X6Y3(w2)Z}
■ {X(2w2)Y(wl>2Z}
where
X = a+2(»i,l)6, Y = 263c, Z = 4c5d.
And as a , 6 , c , d cannot conversely be expressed in terms of X, Y, Z alone, the
lefthand member of the identity, that is, the continuant, can only be made to appear
as a function of X, Y, Z and one of the four a , b , c , d. Consequently, supposing
this to be done, and thereafter all terms involving X, Y, Z deleted, we shall obtain
a continuant which not only vanishes but which can be viewed as having n vanishing
factors.
(11) To obtain this nilfactor continuant there is, however, a better method. For,
as it is the special case where X , Y , Z vanish, it must be obtainable by putting
4c = bd ,
or, therefore, by putting
2b = Sc=—d,
4
2(« !)/>= (« 1)
15
d =
15
Doing this we find from § 9
c =  e,
6 '
h = e,
a =  (7i  l)2e .
/m
1
2m l 6 '
y m
1
2m l 6 '
A
2{n + 2(rol)»l} .
"■m
(2m3)(2ml)
and have the following theorem : — The value of any continuant of the form spoken
of in § 9 is not altered by adding to its matrix the matrix of the continuant
2
■(rel)e 2(ral)e
we
^g(n+l)fl (ft 2)1*
1
( w + 1 )4 7?U» + 7 ) r ' (w3)le
(w + 2)lfl ~(n+\7)e
5 n<
(VIII)
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 351
(12) There is a special case of the theorem (VII) in § 9 which deserves particular
attention, viz. the case where o vanishes, i.e. where 5d = 2b + c. In this case the .s ,th
factor
= {a + 2(nl)b}  2(sl){2&3c} + (n s + 2)(s  l){2b 3c}
= {a + 2(nl)b} + (ns)(sl)(263c);
and the (n — s + l) fch factor
= {a+2(nl)b}  2(ns){2b3c} + (s + l)(n«){26  3c}
= {d + 2(»l)6} + (ws)(sl)(263c).
This means that tuAew 5d = 2b + c in the continuant of § 9 £Ae s th factor from the
beginning is the same as the s th factor from the end, and consequently that an even
ordered continuant of this hind is a square. . . . . . (IX)
(13) The question of the generalisation of the theorem of § 3 may be investigated
in a manner perfectly similar to that followed in the preceding paragraphs with regard
to the theorem of § 2. The essential point of difference is to be found in the new
set of columnmultipliers, which are now all of the form C r+2s ,._ x instead of fj r +ii,i»\'
It will suffice merely to enunciate the results. The first is —
If the continuant
 a (ral)ft •
(n + 2)y l a + p (n2)fi 2 . ....
(w + 3)y 2 a + q (»3)0j ....
(n + 4)y 4 a+r ....
be resolvable into linear factors by means of the set of columnmultipliers
1 51 S 4 K C
1, 4, 10, 20, Cr+2,3
1, 6, 21, C r+4 , 5
1 > 8 > C r+6 ,
£/jeri (l) every four consecutive /3'.s are connected by a linear relation, viz.
JKft 2ft + 00= 9(&2& + /3 4 ),
5(/3 2 2p 1 3 + &) = ll(/3 3 2# 4 + /? 5 ),
£/ms making all the /3's expressible in terms of any three ; (2) all the y's are expressible
in terms of the same three fi's because of the fact that for all values of m (3 m + y m =
— 9j8j + 25/3 2 — 14/8 3 ; and (3) p, q, r, . . . are also so expressible because of the mode
of removing the first factor from the continuant. ..... (X)
It should be noted that the linear relation connecting the first four consecutive 3's
is that which in § 8 connects the second four, — that, in fact, the relation here is
(2r + l)\/3 r  2/3 r+] + /3 r+2 } = (2r + 7){/?, +1  2ft +2 + ft
r+3j
whereas in § 8 it is
(2r  1){/>V  2/3,. +1 + fS r+2 } = (2r + 5){/3 r+1  2/3 r+2 + /3 r+3 } •
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 13). 52
352
DR THOMAS MUIR ON
Further, there is a similar difference in the expressions for o : here the expression is
9ft + 25/3 2 14/3 3 ,
whereas in § 8 when given in terms of /3 2 , /3 g , /3 4 it is
9ft + 25j8 B H£ 4 .
The second theorem is — The continuant
A, (Ml)^
(» + 2)yi A 2 (»2)ftj
(«+3)y 2 A 3 (to — 3)jQ 3
(m + 4) 73 A 4
is resolvable into linear factors if
the s th factor being
(m  l)(m 2) „ j
A. & <»i)« + S<2,£t) 7d '
ym=&+(m + 2)c _ (^3)(^2) , 7ri;
' v 7 2(2m +1)
A w = a  2(m 2  1 )e + ^4^(2m 2 + 2 + 5n) • Id ,
a + 2(7i  2s +\)b 2(«  l)(2n  2s + 3)c
+ («s + 4)(.sl) 7d,
or
{a + 2(«l)6}2(sl){265c}(ns + 4)(sl){4c7^}. . . . (XI)
An immediate deduction from this is that when 14d = 2b + 3c the 8 th factor is
a + 2(n  1)6 + }(»  »)(a  1)(26  5c)
and is the same as the 8 th factor from the end, so that when in addition n is even the
continuant is a square. ....... (XII)
The third theorem is — The value of any continuant of the form referred to in § 3
is not altered by adding to its matrix the matrix of the continuant
2 1
 ^(n + 2)e
35
(m  7)e
1
(» 2)e
 J(«+3)e _ jL(ra17)e i(n3)«
5 57 7
1
'n + i)e
■■9
(» 31)e .
XIII)
(14) The corresponding theorems for the set of columnmultipliers
1; 1) 1; 1 > 1 j J ^il,
1, 2, 3, 4, , C,.,,
1.4 CU 9
will be found in the Trans. S. Afr. Philos. Soc. referred to above.
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
353
When the first theorem there given is attempted to be generalised in the manner
employed in the present paper the following is the result : —
The continuant of the u th order ivhose main diagonal is
a, a + (bc)(n2)y, a + 2(b c)  2(w  4)y , a + 3(&c)  3(n b)y , ....
and ivhose minor diagonals are
(n\)b, (n2)(b + y), („3)(6 + 2y), . . . .
2{cy), 3(c2y), ....
is equal to the product of the n factors
{a + (nl)b\
■ {a + (n2)bc+l ■ (n2)y}
■ {a + (n  3)6  2c + 2 • (n  3)y }
• { a  (n  1 )c)
(XIV)
It is seen to degenerate into the original theorem when y is put equal to 0. If, how
ever, we write X in it for the halfsum of the first and last factors, and Y for b + c ,
the factors may be written
X + i(nl)T,
X + J(ra3)Y+I (n2)y,
X + (w5)Y + 2(«3) r ,
Xl(«l)Y;
thus showing that four variables are not necessary for the expression of the identity.
An easier way of reaching the same result is to put
a = a(n 1)£ ,
b = fS + $,
c=P£,
when it will be found that .£ appears in the continuant but not in its factors ; and
when there are consequently obtained at one and the same time the case of theorem
(XIV) where b = c, and an expression for the corresponding nilfactor continuant.
(15) We have thus in all at present three sets of columnmultipliers, each of which
has associated with it a linearly resolvable continuant of the form
7i
Pi
a + p
y2
a + q
7s
a + r
Other sets will doubtless be discovered, as the only difficulty is the devising of a set
which will not lead to unreasonably complicated expressions for the elements of the
continuant. In all cases if
fc 1 ) fe 2 ' f S ' ' • " • ' =«!
354 DR THOMAS MULR ON
be the diagonal adjacent to the diagonal of units in the set, — that is to say, if the
set be
1 , £ ,
1 , l 2 ,
— and if we denote the elements of the resolvable continuant by their placenames (1, 1) ,
(1,2), . . . , then the factors of the continuant are
{(1,1) +£■(!, 2)}
■ {(2 s 2)£.(l,2) + ! 2 .(2,3)}
.{(3,3) 2 .(2,3) + 4.(3,4)}
{(n.fO^fnl.n)} (XV)
A scrutiny of the procedure connected with the removal of any factor makes this
evident. For, firstly, when s — 1 factors have been removed, the residual determinant
has for its first column the line of columnmultipliers last used, viz.
1, f_i, ;
secondly, this determinant when reduced to the next lowest order has
(s,s)L_ 1 (sl,s)
for the first element of its diagonal ; thirdly, the employment of the next line of
columnmultipliers, viz.
changes the said element into
(s , s)  41 (81,8) + Us ,s + l);
and this, in virtue of the character of the process, is the next factor ready for removal.
It may be noted in corroboration of § 6 that the sum of the factors thus expressed is
(l,l) + (2,2) + (S,3)+ . . . + (n,n).
(10) Observing from the foregoing that
(n ,n)  i n i(n  1 , «)
is the last factor, we have suggested to ourselves the obtaining of the factors in the
reverse order by the use of a set of roiomultipliers, the first operation being
row,  £„_! • row,,.! + . . . .
An interesting result is thus reached, viz., that corresponding to each set of column
multipliers for the resolution of a continuant there is an equally effective set of row
multipliers.
Thus returning to the continuant of § 2 and performing the operation
row 5  8 row 4 + 28 row 3  56 row 2 + 35 tow 1
we find we can remove the factor a — 8b — 8c, and write its cofactor in the form
a 24:1
5(6 + c) a 2c 3(6 c)
6(6 + 2c) rt8r: 2(6  2c)
7(6 + 3c) a 18c 1(6 3c)
35 56 28 8 1
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
355
a
5(b + c)
and thence in the form
246
a  2c 3(6  c)
6(6+ 2c) a 8c 2(6 2c)
35(6 3c) 56(6 3c) 21(6 5c) a + 86 42c
Similarly the operation
row 4  6 row 3 + 15ro\v 2 — lOroWj
now enables us to remove the factor a — 4b— 1 8c ; and the operations
row 3 — 4 row 2 + 3 rowj ,
ro\v 2  1 rowj
the remaining factors. The set of columnmultipliers
1, 1, 1, 1, 1
1, 4, 9, 16
1, 6, 20
1, 8
1
is thus equivalent to the set of rowmultipliers
1 ,  8 , 28 ,  56 , 35
1, 6, 15,10
1, 4, 3
1, 1
1 .
(17) The general result is that the table of rowmultipliers suitable for the
resolution of the continuant of § 2 is
1 ) — Cjn2, l , ^2(12, 2 J — ^'n2, 3 I
1 ) — ^"2n—i, 1 ) C^i4, 2 )
> ( — )" f C. 2 , t _2, n_l
(XVI)
Similarly it is found after a little investigation that the table of rowmultipliers
suitable for the resolution of the continuant of § 3 is
2 2 2
1 ,  y(»  l)O lB _ li o , y(ra  2)C,„_ 1 , 1 ,  — (n  3)C 2n _ 1 ,, 2 ,
1 , hn 2)C 2B _ 3 . 01 , > 3 ) C ^>
. (XVII)
the general form of the multiplier being 2^C 2r+liS .
Lastly, the table of rowmultipliers suitable for the resolution of the continuant oj
§ 13 is
1 > "Uii,n "jii,!j — ^h1,3) ~1
1 , C_*i. C„_, a ,  • (XVIII)
that is to say, may be got by a rearrangement of the columnmultipliers : for
356
DR THOMAS MTJIR ON
example, in the case of the 5 th order the equivalent tables of columnmultipliers and
row multipliers are
1,1,1,1,1 1,4, 6,4, 1
1,2,3,4 1,3, 3,1
1,3,6 1,2,1
1, 4 1, 1
1, 1.
(18) There falls now to be noted a set of theorems regarding resolvable continuants
of a totally different form but connected with and derivable from those of §§ 2, 3, 13.
If in any one of these latter theorems we put for the element in the place (l, 1) ,
the continuant is expressible as the negative product of the elements in the places
(1, 2), (2, 1), and a continuant of the lower order n — 2: further, one of the said
elements is contained in the first factor of the original continuant and the other in the
last factor : in this way, therefore, the resolution of the new continuant of order n — 2 is
secured. Thus, taking the fiveline continuant dealt with in § 2 and putting a = we
obtain
■456(6 cj 8c 2(6 + 2c)
7(6 3c) 18c 1.(6 + 8c)
8(6 4c) 32c
= 86(46 + 14c)(20c)(  4b + 18c)(  86 + 8c) ,
and therefore
8e 2(6 + 2c)
7(6 3c) 18c l(6 + 3c)
8(6  4c) 32c
The general theorems thus obtained are
(46 + 14c) 20c (46 + 18c).
A 1 (»l)/8j
(?j + 4) 7l A 2
(n + 5)y 2
(n  2)&
= 2 n 1 ■ I 2(nl)62(2» + l)c+l.(n + 2)5d
■it + 21 j
{2(n  3)6  4(2ra  l)c + 2(» + l)5<*}
{2(n  4)6  6(2n  3)c + 3n5d}
(XIX)
if
« z. / , i\ , !«(ffl+l) c , i , , , n\ (m+ 2)(m + 3) K ,
K = b  {m + 1), + g^L—JW , y m = I + (« + 2 )C  2(2 J +3) 5,/ .
and
(»!)&
A,„ = (m + 1) 2
(rc+6)y A,
(n + 7)y.
(n2)&
A.
_ 9 , 2m(m + 2) + 6 + 3?* , , \
(2m+l)(2m + 3) ' J '
= 4 ra+  1 •  2(»l)62(2rc + 3)c + l(re + 4).7<2
{2(n  3)6  4(2w + l)c + 2(ra + 8).7d}
{2(«  5)6  6(2n  l)c + 3(ra + 2)7d}
/
(XX)
if
ri i i , i\ , rn(v) + l) 7 . .. (w + 4)(m + 5) „ ,
p,„ = w — (vm + l)c + — i ' la , y.„, = 6 + (?». + 4)c  ^ ; — ^ — — — ' til ,
H V ' 2 2m + 5) ' 7 '" K ' 2(2m + 5)
and
A„, = (to+1)(to + 3) 2c +
2wi 2 + 8to + 10 + 5m
(2m + 3)(2n» + 5)
7/
CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
357
2(b c  ray + 2y) In  1 )(b + 2y)
3(c2z) (36cray+4y) (ra2)(6+3y)
4(c3y) 4(6cray + 6y)
(XXI)
= (ra+l).{ra( i 8 + y)^c}{(ral)(6 + 2y)2c}{(ra2)( / Q+3y)3c}
Of these three results it would be interesting to obtain independent proofs.
(19) In a paper of Heine's on Lame's functions (Crelle's Journ., lvi. pp. 7799)
there occurs the continued fraction
'V'i
2 C ft C<,C S
Z — C, — r,
where the t's are m in number with the values
\m(m+l) , {(m  l)(m + 2) , J(m2)(m + 3), . . . . , \\2m,
the fractional factor being £ in every case except the first. From extraneous considera
tions the value of the continued fraction was given
z( 22 2 )(z4 2 ) . . .
for m even
and
(2  to 2 )
( Z  22 ).,,.( Z ^) f ° rm0dd
but the author added, " Einen directen Beweis fur diese Summirung des Kettenbruches
habe ich noch nicht aufgefunden." This, however, is readily obtained by writing the
value of the continued fraction as the quotient of two determinants, viz.
z c,
h
'■'„
"0
Ci Z Cj — C.) f 2
C 3 Z ~ C 3 ~ r 4
Z  C,  C
and then using two of the foregoing theorems. Thus, taking the case where m = 6,
and where therefore the c's are
21, 10, 9, 7i, 5J, 3,
we have to evaluate the quotient
221 21
10 z19
71 213 51
2 19 9
1
3 23
7i 2  13 5
I . . 3 23
the dividend of which, changed into
221 63i
4(3^1) z21 + 21 2 2.(31+1)
5(312) z21 + 22 2 l(3J + 2)
6(3£3) 221 + 23 2
358 DR THOMAS MUIR ON CONTINUANTS RESOLVABLE INTO LINEAR FACTORS.
is seen to be one of the simplest cases of the determinant of § 2, and thus to have for
its value
g(a4)(216)(z36>.
In like manner the divisor may be written
z19 24J
3(4£2) 219 + 6 1(4£1)
4(4£3) z19 + 16
and is then recognised to be a very special case of the continuant of § 3, and therefore
to be equal to
(*l)(«9)(«25).
The continued fraction in question is consequently equal to
2(34)(216)(z36)
(2l)(29)(225) '
( 359 )
XIV. — The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D.
Falconer, M.A., B.Sc. Communicated by Professor James Geikie, LL.D.,
D.C.L., F.R.S. (With a Map.)
(Read December 5, 1904. MS. received same date. Issued separately June 9, 1905.)
CONTENTS.
PAGE PAGE
Previous Literature 359 The Fourth Volcanic Zone, or The Hilderston and
Introduction 359 Hiltly Lavas 362
The Houston Coal 360 j The Index Limestone and the Bo'ness Lavas . . 363
The First Volcanic Zone, or The Brox Burn Ash . 360 The Fifth Volcanic Zone, or The Kipps and Bishop
The Second Volcanic Zone, or The Longmuir and brae Lavas 364
Riccarton Lavas 361 The Dykeneuk and Castlecary Limestones . . 365
The Third Volcanic Zone, or The Kirkton and Hill ' The Volcanic Necks 365
house Lavas 362 The Intrusive Rocks 365
The Hurlet Limestone 362 General Results 366
Previous Literature.
Geological Survey, sheet 32, 1859 ; revised ed. 1892.
sheet 31, 1875.
Geology of the Neighbourhood of Edinburgh ; Memoir of sheet 32, 1861 ; Memoir of sheet 31,
1879.
H. M. Cadell, " The Geology of the Oil Shalefields of the Lothians," Trans. Ed. Geol. Soc, 1901.
"The Volcanic Rocks of Bo'ness," Trans. Ed. Geol. Soc, 18801.
in Trans, of Inst, of Mining Engineers, Glasgow Meeting, 1901.
Introduction.
The Bathgate and Linlithgow Hills occupy a welldefined belt of rising ground
stretching S.S.W.N.N.E. from Bathgate to Bo'ness, and included in the marginal
portions of sheets 31 and 32 of the 1in. maps of the Ordnance and Geological Surveys.
Throughout the range the steeper slopes face the west, while the eastern flanks are
deeply buried in drift. The glaciated contours are well seen from the east, and the
skyline is in several places deeply indented by glacial grooves. The physical
geography is throughout intimately dependent upon the geological structure, but the
latter is simplicity itself when contrasted with the complicated structure of the shale
fields to the east. Alternating zones of volcanic and sedimentary rocks strike parallel
with the direction of elongation of the range from Bathgate to Bonnytoun Hill, and
these are cut by a later connected series of dykes and sills of intrusive igneous rock.
The latter, as a rule, are more resistant than the lavas, and form the more prominent
features of the landscape, while the sedimentary intercalations may frequently be
traced, even where no rock is visible, by the troughlike depressions which have been
hollowed out of them between the zones of lava.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 14). 53
360 MR J. D. FALCONER ON THE
As convenient boundary lines for the study of these hills, I have chosen as lower
limit the final western outcrop of the Houston Coal from Drumcross to Champfleurie, and
thence a line drawn directly north to the shore of the Firth at Stacks, and as upper limit
the outcrop of the Castlecary Limestone. The accompanying map represents on a
reduced scale the 6in. sheets ix. N.E., v. S.E., v. N.E, i. S.E., i. N.E. of Linlithgowshire,
with the eastern halves of ix. N.W., v. S.W., v. N.W., i. S.W., i. N.W. of the same
county. I have thought it unnecessary to continue the map to the south of the
Bathgate Railway, beyond which the volcanic series rapidly runs out, while the country
is so much covered with drift that little rock is visible. The geology below the
Houston Coal and above the Castlecary Limestone is simply sketched in from the
Survey sheets, and has not been personally verified in detail.
The Houston Coal.
The outcrop of the Houston Coal, crossed by numerous dip faults, has been well
proved from Deans to Drumcrosshall. Thence it strikes north by west to the neighbour
hood of Blackcraig and West Binny, having been worked many years ago at both of
these places. In the vicinity of Ochiltree Mill the outcrop is uncertain, the ground
being much faulted and pierced by many intrusions. North of Peace Knowe, however,
it reappears and strikes north by west to the Haugh Burn fault, by which it is shifted to
the east beyond the limits of the present map.
Between the Houston Coal and the first volcanic zone come the Houston shales
and marls, with thin sandstones and occasional beds of ash and agglomerate. Good
sections are found in the Mains Burn and the Brox Burn and its tributaries.
The First Volcanic Zone, or The Brox Burn Ash.
This zone can be traced, with interruptions, from Drumcrosshall to the Haugh
Burn. It consists throughout of stratified volcanic ash, varying much in texture and
usually green in colour, but weathering yellow or brown at Chapelhill and Bankhead.
Interbedded lava is nowhere found in this zone, although in places the compact ash
weathers spheroidally, and presents a deceptive resemblance to a decomposing
crystalline igneous rock. Characteristic sections are found in the Brox Burn and its
tributary to the south. On account of the absence of exposures, this zone cannot be
traced to the east of the Longmuir plantation. It reappears, however, on the same
horizon on the Riccarton road, immediately to the north.
Between this ash and the second volcanic zone there appear some thinly bedded
sandstones and shales, frequently ashy themselves, and interstratified with thin bands
of ash. These are well seen in the Brox Burn and in a streamlet in the northern angle
of the Longmuir plantation.
IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 361
The Second Volcanic Zone, or TJie Longmuir and Riccarton Lavas.
This zone is taken to include two apparently distinct groups of lavas. The
lower or Longmuir group extends from a point a little to the east of Broomyknowes,
through the Longmuir and Balditop plantations, to Drumcross. The upper or
Riccarton group is of greater thickness but of less extent, stretching only from the
Riccarton Burn to the Rigghead plantation, a little to the south of Tartraven. The
two groups are separated by a series of sandstones, shales, and thin ashbeds, well seen
in the streamlets and quarries on the northern slope of the Riccarton Hills. Petro
graphically, the Longmuir lavas are throughout finegrained olivinebasalts, while the
Riccarton lavas can be separated into a lower zone of coarsegrained olivinedolerites #
extending the full length of the group, and an upper zone of olivinebasalts stretching
from the Riccarton Burn to North Mains.
This subdivision of the zone is everywhere apparent to the north of the Mains
Burn, but to the south so little rock is exposed that the lines are to some extent
hypothetical. The sedimentary intercalation between the two groups of lavas cannot
be traced by means of actual exposures of sandstones and shales. In consequence,
more than usual reliance must be placed upon the prolongation to the south of the
petrographical variations established above. The coarsegrained olivinedolerites of
the lower zone of the Riccarton lavas are nowhere found to the south of the Rioahead
plantation, beyond which the exposures are all of olivinebasalts similar to the Long
muir lavas to the north. The sedimentary band, apparently reduced in thickness, is
therefore drawn, as nearly as possible, between the basalts and the dolerites. Marginal
sedimentary intercalations are probably numerous throughout this zone. Two such
are shown on the map, one of shales and sandstones found in drains on the farms
of Drumcross and Quarter, the other of sandstones, shales, finegrained green ash, and
agglomerate, exposed in the Brox Burn at the Balditop plantation. The sections
exposed during the construction of the Bangour reservoir indicated considerable dis
turbance of the strata at that point. Another thin bed of sandstone, interstratified
with the lavas, may be seen immediately to the east of the neck in the Riccarton Hills.
This second volcanic zone cannot be definitely traced to the south of the Galabraes
fault, although the section at Starlaw indicates the occurrence of scattered lavaflows.
North of Broomyknowes also the lavas rapidly run out, and their place is taken by
interst ratifications of sandstone, shales, and thin ashbeds, with at least one thin band
of shelly limestone exposed in the burn below Riccarton Mill.
Between the second and third volcanic zones there comes another sedimentary
intercalation, represented by shales and thin ashbeds at Craigs, limestone and shale
at Tartraven, shales and ash in the Riccarton Burn west of Beecraigs, sandstone, shale,
and limestone at Whitebaulks, sandstone at Hillhouse, sandstone and limestone at
* The terras "basalt" and "dolerite" are used throughout to denote macroscopic distinctions, "basalt" implying
a finegrained, compact, and usually porphyritic rock, " dolerite " a coarsegrained rock, not evidently porphyritic.
362 MR J. D. FALCONER ON THE
Carsie Hill, east of Cauldhame, and at Peat Hill, on the north side of the Haugh
Burn fault.
The Third Volcanic Zone, or The Kirkton and Hillhouse Lavas.
This zone includes the Kirkton, Tartraven, and Hillhouse lavas. A very charac
teristic ash, with black matrix and yellow lapilli, lies in several places at the base of
this zone. It is well seen at Whitelaw, Craigs, Whitebaulks, and Hillhouse. The
lavas of this zone can be studied with ease in numerous exposures from Kirkton Mains
and Boghall to The Knock. Porphyritic olivinebasalts predominate, but a thin
band of dolerites strikes N. W. from the Raven Craig. The limestones and accom
panying shales and ash of the east and west Kirkton quarries, so well described in
the Survey Memoir accompanying sheet 32, occur as isolated lenticular patches
between successive lavaflows, and evidently occupy a much higher horizon than the
Tartraven Limestone. A similar intercalation of sandstone may be seen on the eastern
slope of the Knock Hill. From the Knock to Tartraven little rock is visible, but
numerous exposures are found in the Tartraven Hills where the road cuts through a
series of darkblue lustrous limburgitic olivinebasalts. The lavas of this zone probably
run out to the north of the Mains Burn, for in a streamlet to the east of Balvormie
the only representative of the zone is the basal ash noted above. This apparently
swells out by Whitebaulks to Hillhouse, where it dips below a group of coarsegrained
olivinedolerites, which, after suffering displacement by the Haugh Burn fault, runs out
to the north of Parkly Place.
The Hurlet Limestone.
This wellmarked horizon can be traced from Glenbare quarry, east of Bathgate, to
the North Mine quarry on the Tartraven road. For a mile to the north of this point
the outcrop is conjectural, no trace of the limestone being found at the surface. It
reappears, however, in characteristic sections in the Hillhouse and Hiltly quarries.
North of Hiltly the outcrop must be shifted to the east by the Haugh Burn fault, and
probably strikes north from the vicinity of Parkly Place to Linlithgow Poorhouse, and
thence northwest to the shore at Stacks. The Hurlet Limestone is throughout
associated with sandstones, shales, and thin ashbeds ; and detailed descriptions of
sections, formerly better visible than now, may be found in the Survey Memoirs and
Mr Cadell's papers.
To the east of Linlithgow no volcanic rocks are found below the Hurlet Limestone,
within the limits of the present map, and, other than the thick sandstone formerly
quarried at Kingscavil, little rock of any kind is visible.
The Fourth Volcanic Zone, or The Hilderston and Hiltly Lavas.
This zone reaches its greatest thickness immediately to the south of Linlithgow,
but even here the apparent thickness is greater than the actual thickness, on account of
IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 363
the effect of the Haugh Burn fault. Petrographically, the zone is composed of a
number of alternating bands of olivinebasalts and olivinedolerites. Transverse
traverses in the neighbourhood of Clarendon, Hiltly, Wairdlaw, or Hilderston readily
reveal this structure. As a rule, however, these bands cannot be traced far in a north
and south direction. A stratified intercalation of ash and ashy shales is found in
Preston Glen, and another of limestone and shales at Wairdlaw. The basalt overlying
this limestone is noteworthy, both for its platy jointing and for the fact that it is
the only lava throughout the whole volcanic series which contains phenocrysts of
felspar in any abundance. The same rock can be traced on the south side of the
valley at Wairdlaw, but the limestone below is nowhere visible. A line of springs
behind Craigmailing probably marks the prolongation to the south of this sedimentary
intercalation, and indications of its presence are also found on the eastern slope of
Cathlaw Hill. Towards the south interbedded sediments probably become more
abundant. Two bands are exposed in the Mavis Glen, and these can be traced for a
considerable distance to the north and to the south by means of the shallow depressions
between the lavas to which they give rise.
The lavas of this zone cannot be traced to the north of Linlithgow Loch. In all
probability they rapidly run out, and their place appears to be taken in part by a
thick bed of volcanic ash found in a bore in Bonnytoun farm. Farther north the
sandstones, shales, and thin limestones of Carriden probably occupy approximately the
same horizon.
The Index Limestone and the Boness Lavas.
Between the fourth and fifth volcanic zones there occurs in the Bathgate Hills an
important belt of sedimentary rock, which includes the lower Bathgate coals and the
Index limestone. It retains a fairly uniform thickness and character from Bathgate
to Kipps. North of Kipps, however, the thickness gradually increases, and volcanic
material becomes mingled with the sedimentary. On the eastern slope of Cockleroy a
bed of ash appears not far below the probable position of the Index limestone, and at
Kettlestoun finegrained volcanic mudstones, passing into ashy sandstones and shales,
are found overlying the lavas of the fourth volcanic zone. Between Kettlestoun and
Linlithgow Bridge the only exposure is in the river Avon at the railway viaduct, where
a very vesicular basalt lies a few feet below the Index limestone. The journals of
bores quoted by Mr Cadell seem to indicate that a considerable proportion of the rock
below the glacial gravels of this district is of volcanic origin. This change in the
character of the strata makes it very doubtful whether the Kipps coals and the Index
limestone are continued across Cockleroy to Kettlestoun. It is quite possible, however,
that the coals do exist, but almost certainly in an attenuated form, of no commercial
value, and much destroyed by intrusive rock.
North of the Edinburgh and Glasgow Railway the Index limestone is repeatedly
exposed in the river Avon, while the sedimentary zone, as a whole, opens out rapidly
364 MR J. D. FALCONER ON THE
to include the Bo'ness coalfield and its intercalated volcanic rocks. In one sense, there
fore, since mere thickness of strata is in this case no index to rate of deposition, the
whole of the Bo'ness coalfield may be considered the equivalent of the lower Bathgate
coalfield. On the other hand, from the position of the igneous material where it first
appears at Coekleroy and Linlithgow Bridge, the lower Bo'ness coals might possibly
be considered more nearly the equivalent of the lower Bathgate coals in point of time.
Further, the lavas of the Bo'ness coalfield apparently form a group by themselves
entirely distinct from the volanic zones of the Bathgate Hills to the south. It is highly
probable, as Mr Cadell has suggested, that the conditions of sedimentation were entirely
different on either side of a volcanic orifice somewhere in the vicinity of Little Mill.
It is unnecessary to describe in detail the stratigraphy of the Bo'ness coalfield.
That has been admirably done already by Mr Cadell, and the Little Mill district alone
still remains more or less of a puzzle. Petrographically, the lavas of Bonnytoun Hill,
south of the Roman road, can be readily subdivided into three zones — a lower zone of
coarsegrained olivinedolerites between the Red Coal and the Wandering Coal ; a middle
zone of porphyritic olivine basalts between the Wandering Coal and the Western Main
Coal ; and an upper zone of coarsegrained dolerites between the Western Main Coal and
the Muirhouse coals. The middle zone alone can be traced below the ash of Little Mill
to Pepper Hill and Linlithgow Bridge. To the north the lower zone can be traced
continuously to the Bonhead fault, but the two upper zones, north of the Roman road,
apparently pass into finergrained doleritic basalts, which persist throughout the
remainder of the coalfield. The rock exposed above Bonsyde is similar to the lavas of
the middle zone to the west, and is probably a displaced portion of the other lavas
of the hill.
The Fifth Volcanic Zone, or The Kipps and Bishopbrae Lavas.
This zone lies a short distance above the Index limestone, and may be traced, with
interruptions, from Linlithgow to Bathgate. Between the Avon Paper Mills and the
vicinity of Coekleroy, where this zone reaches its greatest thickness, the only ex
posure is found in the Cauld Burn at East Belsyde. It is highly probable, however,
that this zone is continuous throughout. Petrographically, the lavas belong mostly
to types of olivinebasalt, and limburgitic varieties may be studied with ease in the
neighbourhood of Kipps. Coarsergrained doleritic types occur here towards the
summit of the series, and appear also in the river Avon at Linlithgow.
Above the lavas and below the Dykeneuk limestone evidence of the continuance of
volcanic action is found in Carriber Glen, where a thick series of ashy sandstone and
volcanic mudstones, in places fossiliferous, are exposed in the gorge of the river Avon.
Similar ashy sandstones are found at Threegables, east of Bowden Hill ; and in a
streamlet between Lochcote and Gormyre a bed of ash occurs on approximately the
same horizon, overlaid by a peculiar blue mudstone, which in places much resembles a
decomposed igneous rock.
IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 365
The Dykeneuk and Castlecary Limestones.
The Dykeneuk limestone, though proved in many bores, is seen at the surface in
three exposures only — at Dykeneuk, Woodcockclale, and Carribber. The Castlecary
or Levenseat limestone is exposed at Craigenbuck, at the Birkhill viaduct, the Avon
aqueduct, Carribber, Bowden Hill, and Lochcote. Both limestones are probably con
tinuous across Bowden Hill, though in places cut out by intrusive rock. South of
Bishopbrae their presence beneath the surface has been repeatedly proved in bores.
The strata between the two limestones consist of sandstones and shales, thickest towards
the north and thinning out towards the south, with the effect of bringing the limestones
closer together from Carribber southwards. No trace of volcanic activity is found
above the Dykeneuk limestone.
The Volcanic Necks.
These are found in the eastern part only of the volcanic area. A small neck full
of green ash pierces the Riccarton Hills southeast of Belcraigs. Another, full of coarse
agglomerate, breaks through the stratified ash of the first volcanic zone to the west of
Wester Ochiltree. A group of seven small necks is found in the neighbourhood of
Hiltly and Parkly Place, some filled with coarse agglomerate, and others with fine
grained ash similar to that at the base of the third volcanic zone. The Necks of
Pilgrim's Hill and Carriden are also included in the accompanying map.
The Intrusive Rocks.
The intrusive rocks of this district are readily separated into two groups according
to their microscopical characters, — a smaller group of olivinebasalts and dolerites, and a
larger group of augite diabases, with little or no olivine, but with frequently abundant
hypersthene. An intersertal microlitic or micropegmatitic groundmass is usually present
in the latter, in greater or less abundance. This difference in mineral composition
seems to be most easily explained on the assumption that the two groups are the
products of different periods of igneous activity. The smaller and more basic group
might readily have been produced from the consolidation of igneous material similar to
that which produced the lavas. They may therefore be regarded as more or less
contemporaneous intrusions. The large group, however, is of a more acid character,
and is certainly the product of a later period of igneous activity.
(a) Contemporaneous intrusions. — Small intrusions of olivinebasalt are found in
many of the Necks enumerated above, but, other than these, few contemporaneous intru
sions have been recognised throughout the volcanic zones. A small intrusion of olivine
basalt cuts the ash of the first volcanic zone in the Brox Burn near the Bangour
reservoir. A short dyke of similar material occurs in the third volcanic zone north
366 IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS.
east of Wairdlaw, and another at Haddie's Walls in the Kipps coalfield. The small
dyke piercing the lavacliff to the west of Hiltly is probably of contemporaneous origin.
Much also of the intrusive rock in the vicinity of Ochiltree Mill, as well as the Walton
and Carriden intrusions, may be referred, from petrographical characters, to the first
period of igneous activity.
(b) TJie later intrusions. — These may be subdivided into two groups : —
1. Dykelike intrusions with a vertical or highly inclined junction —
E.g. The Raven Craig, the Knock, the Witch Craig, the Wairdlaw and
Cockleroy intrusions, and the E.W. dykes.
2. Sills or laccolitic intrusions —
E.g. The Kettlestoun Hills, the Belsyde Hills, and the Torphichen
Hills.
The similarity in petrographical type of these intrusions indicates contemporaneity
of origin in late or postcarboniferous times. Strict contemporaneity, however, is not
implied, the intrusions having evidently been inserted in succession during the period.
Those intrusions with a N.S. elongation are probably oldest, being cut, as in the case
of the Raven Craig, by the E.W. faults, along which dykes have usually risen. These
dykes are apparently the feeders of the sills, the lowest and oldest of which are some
times covered by the feeders of the higher and younger. The upper limits of the dykes
themselves were probably irregular, different portions of the same dyke rising to
different levels. The upward termination of one of these dykes is particularly well
seen at Broomyknowes and Belcraigs. This may explain in part the discontinuity of
outcrop of some of these dykes when traced from east to west. In other cases, how
ever, as in the Parkly Craigs dyke, the different portions seem to run out, the dyke
being continued on a parallel line a few yards to the north or south.
General Results.
1. The lavas of the Bathgate Hills are olivinebearing from base to summit of the
series, and are pierced by a few contemporaneous intrusions of similar material.
2. The Bo'ness lavas form a group entirely distinct from the lavazones of the
Bathgate Hills.
3. The volcanic zones are crossed by a laterconnected series of dykes and sills,
probably of Palaeozoic age.
The detailed results of the microscopical and chemical analysis of the rocks are
reserved for a future paper.
( 367 )
XV. — On a New Family and Twelve New Species of Rotifera of the Order
Bdelloida, collected by the Lake Survey. By James Murray. Communicated
by Sir John Murray, K.C.B., F.R.S. (With Seven Plates.)
(MS. received January 13, 1905. Read January 23, 1905. Issued separately March 3, 1905.)
Introduction.
The new species here described were found in the course of the work of the Lake
Survey on Loch Ness and other Highland Lochs. Half of the number were found in
lakes, though they are not exclusively lacustrine, three in ponds, two from moss growing
on the shores of Loch Ness, and one in a stream running into Loch Ness.
Structure. — A short account of the structure of a typical Bdelloid will be necessary,
in order to render intelligible the terms used in the descriptions. A Bdelloid is a
Rotifer which can creep like a leech or caterpillar. The body is segmented, and
consists of head, neck, trunk, and foot. The head, neck, and foot are telescopic, and
can be completely withdrawn into the trunk. The normal number of segments is
sixteen, but there may be more or less, the variation being chiefly in the foot. The
head consists of three segments, the neck of three, the trunk of six. It is believed that
the number of segments in each of those portions of the body is invariable, but two or
more segments may be so united as to be indistinguishable. The foot is more variable,
the number of segments, normally four, varying from one to six. Beginning at the
anterior end, the first and second segments of the head form the rostrum. The first
has an inverted tip, from which rise the two rostral lamellae, numerous motile cilia, and
sometimes larger tactile setae. The third segment is the oral, and bears the mouth,
and the corona when present. The first cervical bears the antenna, and frequently a
number of prominences. The second and third cervical have no appendages. The first,
second, third, and fourth segments of the trunk are called the central, and form the
broadest part of the body. The next two segments of the trunk, the preanal and the
anal, together form the rump, which is generally clearly marked off both from the
central part of the trunk and from the foot. At the end of the anal segment is the
anus. The segments beyond the anus constitute the foot. The first joint of the foot
commonly has the skin on the dorsal surface thickened, and often bears a rounded boss
or other processes. The penultimate joint bears the spurs. The last joint bears the
toes, or the perforate disc which takes their place. The segmentation is superficial, and
affects only the skin. When the animal is fully extended, the various organs usually
occupy definite segments, though the arrangement is not invariable. The brain,
generally somewhat triangular, occupies the second cervical, but when large may extend
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO 15). 54
368 MR JAMES MURRAY ON A NEW FAMILY
through all three cervical segments. The eyes, when present, are in the rostrum, near
its tip, or on the back of the brain. The mastax, containing the jaws with their teeth,
is in the third cervical or first central. The stomach extends through all four central
segments, which also contain the ovaries and the eggs or young. The preanal contains
the intestine, the anal the contractile cloaca. The footglands occupy all the joints of
the foot, and may extend into the anal, or even into the preanal. The toes are three
or four in number, or they are united to form a disc, which is perforate with pores for
the passage of the mucus.
The corona of the Philodinadae consists mainly of two nearly circular discs, borne
on pedicels. The pHncipal wreath borders the discs. The secondary wreath runs
round the bases of the pedicels from back to front, and merges in the cilia of the mouth.
Near the centre of each disc is in many species a seta, or pencil of setae, or several short
motile cilia, which usually rise from a small papilla or a larger process. The lower lip
is the central portion of the under side of the mouth, and is shaped like a V. The upper lip
is the space between the bases of the pedicels and the front of the rostrum. Its form is
very characteristic for each species. Its most important structures are two folds of skin,
which continue those prominences at the sides of the mouth known as the collar.
These folds run round the bases of the pedicels, close to the secondary wreath, and may
meet in the middle line just in front of the rostrum, or may terminate at some distance
apart, in processes of various form. In the middle line, between the pedicels, is often
found a peglike process, known as a ligule. The watervascular system, with its
vibratile tags, usually about six pairs, is difficult to observe. The number of pairs of
tags seen is always noted, though this may not be the full number present.
The skin may be smooth and hyaline, stippled with pellucid dots, viscous, papillose,
or variously warted or spiny. The back and sides of the trunk are longitudinally
plicate. The ventral side is obscurely transversely plicate. In a few species the
ventral transverse folds are numerous and deep.
Habits. — The great majority of the known species are free and independent animals.
None are truly pelagic. Even those which are in the habit of swimming only do so for
short distances, and in the shelter afforded by mosses and other water plants. Those
which do not swim creep in caterpillar fashion on the plants among which they live or
on the mud of ponds. When feeding they anchor themselves by the foot. The
Adinetadse and some Callidince can also glide forward by the action of certain cilia —
in the Adinetadse those of the corona, in the Callidince those of the rostrum. Several
species secrete protective cases ; others accumulate irregular tubes of d6bris.
Parasitism. — A number of species are ectoparasites upon other animals. None are
internal parasites or feed upon their host. They are commensals or messmates. They
attach themselves by the foot to some crustacean, insect larva, or other animal. They
seem to desire from their host only protection, conveyance from place to place, possibly
a share of food. Asellus is a favourite host, and often carries several species together.
All the Bdelloids known to me which have taken to the parasitic mode of life are large
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 369
animals. They are distinguished by bulk of body, long and powerful foot, large strong
spurs, and usually ample corona. The life agrees with them. Signs of degeneration
are not lacking, however. Their affinities are with genera which normally possess eyes,
but most of these parasites are blind, and have smaller brains than are possessed by free
living species of the same genera.
Symbiosis. — A number of Bdelloids have the foot of a peculiar type, in which there
are no distinct toes, their place being taken by a disc, which is perforated by numerous
pores for the passage of the mucus. It has been suggested that the species having this
kind of foot live in symbiotic relationship with certain Hepatics, such as Frullania,
which have some of their leaves in the form of little pitchers. The suggestion is plaus
ible, inasmuch as such Hepatics are seldom found without the pitchers occupied by the
Rotifers. There is, however, something to be said against the belief that the relation is
one of symbiosis. There is no evidence of any advantage to the plants, though it has
been guessed that the animals are in some way beneficial to them, and that to obtain
this benefit the pitchers have been developed as an attraction to Rotifers. It is more
probable that the pitchers of the Hepatics serve the sole purpose of retaining moisture,
and that many species of Rotifers have found and taken advantage of those little
reservoirs. In that case there is no symbiosis, only a mild form of parasitism. Bdelloids
having the discoid foot are not only found on Hepatics. They abound in many other
situations. Nor are they the only Bdelloids which frequent Frullania cups. Many
species with the ordinary foot are commonly found there. It might be supposed that a
Bdelloid would have less need of a strongly adhesive foot in the shelter of a Frullania
pitcher than in many more exposed situations. The discoid foot is an advantage to a
species in any situation, and it is to be noted that all the species having it are large,
powerful animals.
Two species having the discoid foot are here described. One is from a lake, the
other from a pond, and neither has yet been found on Hepatics.
Formation of foodpellets. — Four of the species described in this paper belong to
that section of the genus Callidina in which the food is moulded in the oesophagus into
pellets. All the animals having this characteristic agree in many other points of
structure, such as the small size of the corona, and form a very natural group. Most of
them have the neck very long and the gullet correspondingly elongated, and forming a
large loop when the neck is contracted. The pellets differ greatly in consistence in
different species. Some are loosely put together, and quickly disintegrate when passed
into the stomach. In some species they seem to be mixed, while in the oesophagus, with
something which gives them coherence. Such pellets maintain their size and form
unchanged during the whole of their passage through the alimentary canal, and are
finally voided entire. When first passed into the stomach they are granular, and often
dark from the admixture of food particles. They gradually lose the granular character
as they move through the stomach, till when passed out they are clear spheres.
370 MR JAMES MURRAY ON A NEW FAMILY
BDELLOIDA.
All the Bdelloid Rotifera hitherto known have been included in two families,
distinguished by different types of corona. The Philodinadse have the corona divided
into two discs, which bear the primary and secondary wreaths of cilia. The Adinetadse
have no discs, the corona consisting of a flat surface, furred with short cilia, divided by
a nonciliated space in the middle line, which may correspond to the space separating
the discs in the Philodinadse.
An animal discovered in Loch Vennachar in 1902, in the course of the work of the
Lake Survey there, could not be referred to either of the known families. After
prolonged study, continued for more than two years, it is now proposed to constitute a
new family for its reception.
Microdinad^e, n. fam.
No corona, the ciliated alimentary tract ceasing at the mouth ; jaws intermediate
between the ramate type of all other Bdelloida and the malleoramate type of
Melicerta.
The only species at present known is a Philodinoid animal. It resembles the genus
Philodina in general form, in the rostrum, and in having four toes. The absence of
corona would not of itself have justified the establishment of a new family. It might
have been regarded as a degenerate Philodina which had taken to a different mode of
feeding, and lost its corona from disuse. It was only after the peculiar structure of
the jaws was understood that it became evident that the definition of the family
Philadinadse could not be modified to include it.
As now understood, the new family is seen to differ more from the other two families,
of the order than they do from one another. The Adinetadse differ from the
Philodinadse only in the form of the corona, and in the partly adnate rostrum, free
at the tip. They have the same form of jaws and of all other structures.
The Microdinadse differ from both, not only in the lack of corona, but in the shape of
the jaws. It comes nearer to the Philodinadse in the free rostrum and the number of
toes. On the other hand, the form of mouth might more readily be derived from that of
Adineta. In Plate II. are shown heads of Philodinadse (fig. a), Adinetadse (fig. c),
and Microdinadse (fig. 6). They are drawn from the ventral side in order to show the
similar form of lower lip in all. On the same plate are drawn three pairs of jaws : —
fig. d shows the ramate jaws of Philodinadse and Adinetadse, fig. e those of Micro
dina, fig. f those of Melicerta. It will be seen that the jaws of Microdina differ
about as much from the ramate as from the malleoramate type, and sufficiently
approach the latter, in the anterior position of the teeth and the less rigid union of the
various parts, to constitute in some degree a link between the Bdelloida and the Rhizota.
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 371
Microdina, n. gen.
Toes, four. Yolkmass with four nuclei. Gullet very short. Teeth, three or four
on each side, at anterior end of jaws.
The terminal cilia of the gullet, which project a little way and assist in seizing the
food, might be regarded as constituting a rudimentary corona. It is not usual, how
ever, to regard the end of the gullet in a Philodine as part of the corona, that term
being restricted to the discs and the two wreaths. It seems, therefore, more correct to
consider Microdina as having no corona.
The parts of the jaws are movably articulated, not rigidly united as in other
Bdelloids. The rami have large curved processes on the ventral side. The manubrii
may have no loops, or may have from one to three, more or less distinct.
In Philodinadse and Adinetadse the larger teeth cross about the middle of the jaw,
and there are finer strise towards each end. In Microdinadse the large teeth are all at
the anterior end, and fine strise only cross the posterior half of the jaws.
Owing to the shortness of the gullet, the jaws are close to the mouth.
Microdina paradoxa, n. sp. (Plates I. and II.)
Specific characters. — Of moderate size, stout, enlarged at level of mastax and at
posterior part of trunk, contracted between those parts. (Esophagus and large
granular mass connected with it, of a bright crimson colour. Stomach voluminous,
its walls filled with coloured globules. Foot of three joints. Spurs short, stout,
somewhat bottleshaped, a broader basal portion contracting into a narrower apical
portion, obtuse, separated by narrow convex interspace. Toes large and thick, the
ventral pair much larger than the dorsal. Footglands forked. Antenna twojointed,
flattened. Oviparous.
General description. — Greatest length y^ to g^ inch. Always fiddleshaped,
owing to the narrowing between the head and the enlargement of the trunk. The
position of the posterior enlargement varies. When an egg is carried, the broadest part
will be in the second or third central segment. When there is no egg and the ovaries
are undeveloped, the fourth central or preancd may be broadest. The rostrum differs in
no way from that of a typical Philodine. The mouth is small and somewhat trifoliate.
The lower lip is of the Vshape usual in the order. The upper part is obscurely two
lobed. The sides of the mouth are finely longitudinally striate. The whole animal is
sometimes pale rosecolour or purple. More generally it is colourless, except for the
crimson gland and oesophagus, the stomach and the egg. The globules in the stomach
walls vary greatly in colour. They have been seen red, yellow, greenish, orange,
magenta, sienna, or umber. The two last colours are commonest, and are used in the
illustrations. The egg is of a tawny yellow. A clear fleshy mass fills the head
372 MR JAMES MURRAY ON A NEW FAMILY
between the rostrum and the oesophagus. The posterior portion of this mass is, from
its position and its connection with the antenna, regarded as the brain, but its outline
could not be traced. Between the mastax and the stomach are two clear gastric glands,
which meet on the ventral side.
Habits. — Of tireless activity. It creeps without ceasing on the stems of algae and
mosses, feeding all the time. Its mode of feeding is unlike that of any other Bdelloid,
though Adineta resembles it in some respects. A biting action is continually repeated.
In this the rostrum takes part. The food is caught between the rostrum and the
lower lip, and pushed close to the mouth by the bending down of the rostrum. The
cilia of the mouth, working downwards, catch the food that is thus brought near and
sweep it into the gullet. The brush of cilia on the rostrum contributes to the action by
sweeping downwards also, and to some extent compensates for the lack of discs. It was
never seen to pause or rest, as other Bdelloids do occasionally.
The deposition of the egg was on one occasion seen. The animal was fully con
tracted. When the egg was almost completely passed out, the end which still remained
in the aperture was seen to be surrounded by a circlet of clear spherical bodies. Most
of these adhered to the egg when it separated, but a few remained attached to the
aperture (Plate I. fig. d).
Variation. — Only one species of the family is known with certainty. The lack of
corona deprives us of several characters of great service in distinguishing species of
Philodinadge. Various forms of spurs have been seen in Microdinadae, but it is not
yet clear whether any of these belong to distinct species (Plate II. figs, g toj).
The jaws also differ in different examples. While agreeing in general features, the
degree of development of the loops of the manubrium varies greatly in different
individuals. Some show no trace of any loops, while others have three well developed,
two on the outer side and one on the inner, passing behind the teeth.
Habitat. — At the margins of large lakes and of clear hill lochs, also occasionally in
pure running water.
Discovered in Loch Vennachar, 20th May 1902, on the occasion of the visit of the
Scottish Natural History Society, as guests of Sir John Murray ; Loch Ness and
Loch Morar, 1903; hill lochs on Carnahoulin, FortAugustus, 1904; Loch Treig,
December 1904. Very abundant in Loch Vennachar and frequent in Loch Ness.
Philodinadge.
Classification. — The Ehrenbergian division of the Philodinadae into genera dis
tinguished by the presence or absence of eyes, and by the position of the eyes when
present, has long been recognised as artificial. In those genera unrelated species are
brought together, and closely related species are separated. Suggestions for a more
natural classification have been made, notably by Milne, but none have been generally
AND TWELVE NEW SPECIES OF ROTIFER A OF THE ORDER BDELLOIDA. 373
accepted. The number of known species belonging to this family is now so great that
some subdivision of the genera would be desirable, even if those genera were natural.
Many of the new species show the artificial character of the old genera, and render a
revision imperative.
I understand that a revision of the genera is now being prepared by Mr Bryce, who,
from his long experience of the order, is so well qualified to do so. This being so, I
shall here only amend the definitions of the genera Philodina and Callidina so as to
render them more natural. The classification based upon the eyes having proved
defective, other characters of a more reliable nature have been sought. The number of
toes has been suggested by Milne * as a basis for classification. The mode of repro
duction was thought of. It was found that large groups of species agreeing in the
number of toes, also agreed in the mode of reproduction. One or two exceptions,
however, lessen the value of the mode of reproduction as a generic character, and it
must be abandoned in the meantime.
Philodina.
Generic character. — Toes, four. Milne's suggestion is adopted, though it is
recognised that the genus will have to be divided. Thus defined, the genus does not
differ greatly from that of Ehrenberg. All the species having eyes in the neck {i.e.
seated on the brain) are found, with one exception, to have four toes. The main result
of the alteration will be the transfer to Philodina of several species hitherto included in
Callidina.
Callidina.
Generic characters. — Toes, three ; or foot ending in a disc. Normally oviparous.
This definition is simply provisional. It is unsatisfactory, in that it includes a
character, viz., the mode of reproduction, which is not quite invariable. It is only by
using this character that the genus Rotifer could be kept separate.
As formerly distinguished by a single negative character, viz., the absence of eyes,
the genus Callidina became the receptacle for all the homeless wanderers of the family,
till it now includes a host of species, many of which have little affinity one with
another. It is with this genus that a revision of the family will be mainly concerned.
Four of the new species here described belong to that very natural section of the genus
in which the food is moulded into pellets. Two have the discoid ' symbiotic ' foot.
This type of foot might be made the basis of a genus, were it not that it is in some
cases impossible to determine whether there are separate toes or not. It is, moreover,
suspected that the discoid foot may have been independently acquired by unrelated
animals.
* Proc. Phil. Soc. Glasgow, vol. xvii. p. 134, 1886.
374 MR JAMES MURRAY ON A NEW FAMILY
Callidina angusticollis, n. sp. (PL III. figs. 2a to 2k.)
Specific characters. — Small, colourless ; form pitchershaped in lateral view, the
lower lip large, elevated, spoutlike. Discs small, close together, inclined obliquely
towards the mouth. Oral segment elongate, encircled about midway by a series of four
thickenings. Food moulded into pellets. Foot minute, not obviously segmented ;
spurs short, acute, decurved, meeting at base. Dental formula 2/2. Secretes a brown
flask shaped protective case.
General description. — Greatest length ^ to j^ inch when feeding. Head laterally
compressed, elongate from front to back. Discs sloping downward and outward from
middle line as well as forward towards the mouth. Lower lip larger, relatively to the
size of the animal, than in any other species known. Thickenings on oral segment
diagonally placed, as shown in section, fig. 2e. Rostrum of moderate length, with
fairly large lamellae. Antenna of two joints, length equal to f diameter of neck.
Neck with large rounded thickenings at each side of antenna, and ventral thickening.
Neck very long and slender. Gullet correspondingly elongated. Stomach voluminous,
filled with round, clearly outlined pellets of uniform size. These are coherent, and do
not disintegrate in their passage through the alimentary canal. They are voided whole.
No eyes. Reproduction oviparous. Case oval, slightly flattened on ventral side,
pale yellow when young, dark brown when old. Neck of case long, with annular striae,
mouth slightly expanded.
The foot, being apparently useless inside the case, which the animal never seems to
leave, is very small. It can only be seen when the animal is forced out of its case.
No separation of the first and second joints can be distinguished. The rudiments of
toes probably exist, as the spurbearing joint is not closed at its lower end, but they
were not seen.
Habits. — Trusting apparently to the protection afforded by its shell, it is not at all
shy, and usually resumes feeding very soon after being disturbed. When feeding, the
neck is bent backward. Before beginning to feed, the head is often put out and the neck
bent sharply over the edge of the case till the rostrum touches the outside of the case.
The case is believed to be secreted from the skin, but the process has not been observed.
The animal may occupy empty (or even inhabited) shells of Rhizopods, such as Difflugia
or Nebela. Careful examination has always revealed a normal case inside the shell thus
occupied. On one occasion the Callidina was seen in a shell of Difflugia which was
shorter than its case. The projecting neck of the case was viscous, as shown by
adherent matter, and nearly colourless. It had probably been just completed. Old
animals show no viscosity, either of skin or case. The case is thin, smooth, and brown,
and does not adhere to the animal. It is a cleanly animal. The pellets, which at first
contain the food, are eventually passed out as clear spheres. After voiding them it
clears them out by fully contracting its body and rolling about from side to side of the
case till they are forced out through the neck.
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 375
Habitat. — On the leaves of mosses and hepatics in a great variety of situations, in
peat bogs, on the ground, walls, or trees, most frequently seen on Fontinalis growing at
the margins of lakes. First seen in Loch Morar, common about Fort Augustus ; occurs
also in North Uist. Probably widely distributed. Before the animal came under my
notice Mr Bryce had made some studies of it, and suggested the specific name. Dr
Penard has also studied it in Switzerland.
Callidina annulata, n. sp. (Plate III. figs. 3 a to 3/)
Specific characters. — Small, colourless, in lateral view pitchershaped, the lower lip
spoutlike. Discs inclined towards mouth, their surfaces parallel to long axis of body.
Oral segment much elongated, marked by annular plicae, which are stronger towards the
base. First neck segment with similar plicae. Antenna very small, its length equal
to f of the diameter of the neck. Teeth, seven or eight in each jaw. Food moulded
into pellets. Foot short, of three joints. Spurs, short cones, meeting at base. Repro
duction oviparous.
General description. — Length about T ^ inch when feeding. Oral segment twice
as long as broad. Discs reniform, separated by very narrow sulcus. Neck and gullet
very long. Rostrum short and broad, with small lamellae. Stomach large, nearly filling
the trunk, containing clear rounded pellets of uniform size.
Resembling C. angusticollis in size and general form, it may be easily distinguished
from that species by the smaller lower lip, greater forward inclination of the discs,
longer oral segment, with annular plicae and without thickenings, numerous teeth, larger
foot, and lack of protecting case. Some examples carried large oval eggs. Intestine,
glands, and vibratile tags were not observed.
Habits. — Being unable to secrete a case for itself, as is done by C. eremita and
other species, it seeks shelter, like the hermit crab, in the empty shells of other animals.
Shells of Diffiugia, Nebela, and other Rhizopods are commonly occupied. It was first
observed in cases of C. angusticollis, the original occupants of which had died, leaving
only the tough jaws behind. The presence of those jaws, with their pairs of teeth, led
to the two species being confused for some time. It also frequently takes cover in the
pitchers of Frullania and other Hepatics. It is often found creeping about without
protection of any sort, but it has never been seen to feed unless when in a shelter of
some kind. When feeding it is not timid. It may frequently be observed, in detached
pitchers of Frullania, whirling rapidly about, regardless of collisions.
Habitat. — Among aquatic mosses growing in Loch Morar, October 1903, Loch
Ness, 1904. Not confined to lakes. Common on Hepatics, Fort Augustus, Blantyre
Moor.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 15). 55
376 MR JAMES MURRAY ON A NEW FAMILY
Callidina crenata, n. sp. (Plate IV. figs. 6a to 6d.)
Specific characters. — Small, colourless. Trunk and foot papillose. Head and neck
smooth. Neck with a prominence on each side of the antenna. Teeth, seven or eight
in each jaw. Foot of three joints. A crenate boss on first joint. Spurs short, tapering,
acuminate, divergent. Toes, three, Food moulded into pellets. Posterior margin of
preanal segment with a rounded prominence, free from papillae, on each side.
Oviparous.
Description. — Length gV i ncn when fully extended. Rostrum short, with lamellae
of moderate size. About twelve longitudinal folds on the trunk, at equal distances apart,
not fainter dorsally. Papillae not crowning the folds, as in C. aspera, Bryce, but
regularly distributed over the whole surface of the trunk, smaller than in C. aspera,
rounded, without pits or pores, diminishing in size on the foot. Spurs dotted. Toes
short, blunt. Egg elongate, narrowed at anterior end.
This description is incomplete, as the animal was never seen to feed. Seen in the
retracted state the discs are small and close together. The only other species which
moulds the food into pellets, and at the same time has the skin papillose, is C. aspera,
Bryce. From that it is distinguished by the more numerous teeth, smaller papillae,
and preanal processes.
Habits. — Although fairly abundant in several collections, nothing could be learned
as to its habits. All the examples studied were very sluggish in their motions. They
crept about very slowly ; and though some of them were watched for long periods, they
showed no disposition to feed.
Habitat. — Among ground moss and hepatics from the shores of Loch Ness and
elsewhere near Fort Augustus, February 1904, frequent; not yet found anywhere
else.
Callidina pulchra, n. sp. (Plate IV. figs. 5a to 5f. )
Specific characters. — Small, colourless. Trunk very broad, strongly stippled.
Corona narrower than neck or collar, with central setae on discs. First neck segment
with the anterior edge turned outwards like a rim all round. Rostrum short and broad,
with a large brush of long cilia. Teeth, three to five in each jaw. Food moulded into
pellets. Foot short, of three joints. Spurs short, divergent, acuminate. Toes, three.
General description. — Length about y^ inch when creeping, ^q inch when
feeding. Very short and broad. Skin not papillose, but covered with uniform large
clear dots. Trunk longitudinally plicate ; dorsal folds faint, lateral deep. Stomach very
voluminous, filled with large pellets.
Very similar to C. lata, Bryce, to which it is closely related. It agrees with that
species in the breadth of trunk, the shape of the corona, the central setae on the discs,
and the dental formula. It differs in the oval rather than ovate trunk, the stippled skin,
the projecting edge of the first neck segment, and the shorter spurs. The shape of the
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 377
trunk makes it a more elegant animal than C. lata, the great posterior breadth of which
imparts a clumsiness of gait as it moves.
Habits. — In its steady, deliberate motions and mode of feeding it resembles C. lata.
When creeping it goes steadfastly forward, increasing the length of each step by a glid
ing movement produced by the cilia of the rostrum. It feeds for shorter periods than
C. lata.
Habitat. — In ponds near Fort Augustus, February 1904. Found among the sedi
ment obtained by washing aquatic mosses. It was very abundant in some ponds. When
a portion of the sediment was put into a bottle with some water and tightly corked the
animal continued to abound, and increased in numbers for some months, though the
water was never changed.
Callidina muricata, n. sp. (Plate V. figs. 7a to 7h.)
Specific characters. — Of moderate size, narrow. Trunk with strong longitudinal
plicae, covered with low rounded tubercles. Corona narrower than neck. Discs large,
separated by very small interstice. Rostrum narrow, with large lamellae, which project
laterally. Antenna slender, length equal to half diameter of neck. Neck with large
thickenings on each side of antenna. Brain large, elongate ; no eyes. Dental formula
2/2 ; border of jaws crenate. Food not moulded into pellets. Foot short, of four joints.
Spurs slender, tapering, meeting at base, divergent, incurved. Toes, three ; large, taper
ing. Reproduction oviparous.
General description. — Greatest length ^ to j^q inch. Skin of trunk yellowish,
viscous, with little extraneous matter adhering. Stomach large, its walls containing
small darkgreyish globules. Yolkmass with eight nuclei. Egg large, oval. Discs
nearly touching. Border of jaws brown.
The tubercles are of equal size, rounded, and disposed in transverse and longitudinal
rows. They are probably permanent, and not mere hardened secretions as in C.
incrassata, but this is not proven. On the back they are hidden by the deep longitudinal
plicae. The transverse rows, about nine on the trunk, give a false appearance of close
segmentation. The tubercles are more obvious on the ventral side, and all over when
fully retracted. The glands, intestine, and cloaca were normal. Vibratile tags not
seen. Apart from the tubercles, the species may be known by the close approximation
of the large discs and by the caliperlike spurs.
Habits. — Very slow in its motions. It extends itself with studied deliberation, like
Rotifer tardus, and is not often willing to feed. It feeds steadily, but only for a short
time. On all the occasions when it was seen feeding the ventral side was uppermost,
so that the details of the upper lip could not be seen.
Habitat. — In the sediment of ponds, Fort Augustus, January 1904, frequent;
Blantyre Moor.
378 MR JAMES MURRAY ON A NEW FAMILY
Callidina crucicornis, n. sp. (Plate V. figs. 8a to Sg.)
Specific characters. — Large, slender, elongate. Rostrum very long, of two con
spicuous joints, with very large, spreading lamellae. Antenna very small. Brain large ;
no eyes. Jaws relatively very small ; dental formula, 2/2. Stomach voluminous ; food
not moulded into pellets. Foot short, of three joints, very prominent dorsal boss on
first joint. Spurs long, tapering, with distinct shoulder on inner side at base, capable
of being brought together at the points or crossed over one another. Last joint of
foot long, with three very large toes.
General description. — Greatest length g^ to ^ inch. Every part elongate except
the foot. Colour dull yellow or greyish. Longitudinal plicae few, fainter on back.
Salivary glands well developed, one long narrow pair extending beyond the mastax to
the upper part of the stomach. Walls of stomach thick, filled with larger and smaller
dark yellow globules. Intestine oval, its long axis transverse, partly covered in dorsal
view by stomach. Yolkmass large, with eight small nuclei. Space between spurs
straight or convex, according to position of spurs. Terminal toes long, slender, two
jointed. Dorsal toe as long as the others, but usually extended to only half the length.
Footboss pointing backwards.
Owing to its disinclination to feed, the description cannot be completed. In the
retracted state the discs are large and elongate. The species has a superficial
resemblance to Callidina longirostris in the long rostrum and spurs, and also to
Philodina macrostyla and its allies. It is believed to have no close affinity with any
of those species, all of which are viviparous, while this is oviparous. The rostrum
tapers gradually from the oral segment, and is not abruptly narrowed as in C.
longirostris.
Habits. — Although it has been known for more than two years, and has been under
constant observation for nearly one year, and thousands of examples have been carefully
studied by three or four observers, little is known of its habits, as it has never once
been seen to feed. It creeps slowly and deliberately, examining everything it
encounters with its rostrum, which appears to be a very delicate organ of touch. It is
very mobile, and can be bent backwards and forwards and from side to side. The
lamellae, which are only inferior in size to those of C. cornigera, are waved about in
the way characteristic of that species, and which has led to the supposition that they
are organs of smell.
When washed out of the mosses among which it lives, and allowed time to settle
down among the sediment, it is found that it takes up its position, not on the surface
of the sediment, but a little way down in it. The stomach is often seen to be well
filled with food. These facts, together with its disinclination to feed, lead me to
suppose that it may have an aversion to light, and will not feed unless in darkness
If this is so, it may be impossible to complete the description of the head. Against the
AND TWELVE NEW SPECIES OF EOTIFERA OF THE ORDER BDELLOIDA. 379
suggestion is the fact that though it will not feed, it does not appear to be uneasy in
the glare of the microscope lamp.
Habitat.— In lakes and ponds. Bottom of Loch Rannoch, at depth of 9 or 10 feet,
April 1902. Abundant in pond near Fort Augustus, January 1904. It thrives well
in tightly corked bottles, and may continue in them for months.
Callidina armata, n. sp. (Plate VI. figs. 10a to 10/i.)
Specific characters. — Large, massive. Corona broader than trunk. Rostrum short,
broad ; lamellge small. Antenna as long as diameter of neck, clavate. A pair of tooth
like processes close below the mouth. Jaws relatively small, with two teeth on each.
Stomach voluminous, reddish. Foot short, of three joints. A broad rounded fold at
base of first joint. Spurs small, acuminate, incurved and decurved, interspace equal
to diameter of base of spur. Foot ends in round perforated disc.
General description. — Greatest length when creeping, ^ inch. Trunk with few
longitudinal folds, dorsal faint, lateral deeper. All colourless, except alimentary canal.
Food not moulded into pellets. Walls of stomach containing small reddish globules.
Intestine roundish. Footglands of few cells, terminal cell largest. First footjoint
marked with annular striae. Terminal joint long, disc perforated by many pores, ducts
in common sheath. Four pairs of vibratile tags were seen.
Resembling C. symbiotica and allied species in massive build and discoid foot, it
may be distinguished from all other species by the ventral processes below the mouth,
the heavy antenna, and the dorsal fold at the base of the foot.
Habits. — Strong and active, like all the ' symbiotic ' species. As it creeps rapidly
about, the disc is exposed for an instant. It is a steady feeder. The function of the
processes below the mouth could not be gathered from its actions.
Habitat. — On water weeds growing in Loch Ness and the Caledonian Canal at
Fort Augustus ; although abundant during November and December 1903, it was not
again found till December 1904, when it once more became common. The same beds
of weeds, chiefly Myriophyllum and Fontinalis, were frequently examined during the
intervening months without the species being once found. This may indicate that it
has only a short season, though it is unusual for Bdelloids to have any seasonal
limits.
Callidina incrassata, n. sp. (Plate VI. figs. 9a to 9f.)
Specific characters. — Large, stout. Trunk protected by thick plates formed of a
hardened secretion. Rostrum short and very broad, with small ciliate lamellae.
Antenna considerably longer than diameter of neck. Neck with large process at each
side of antenna. Corona as wide as trunk, discs large, interstice equal to half diameter
of disc. Central papillae on discs. Foot very short, of three joints. Spurs small, twice
380 MR JAMES MURRAY ON A NEW FAMILY
as long as broad, acuminate, divergent, obtuse, incurved. Jaws with broad, brown,
pectinate border, dental formula 5/4. Food not moulded into pellets. Oviparous.
General description. — Greatest length £q to ^ inch. Trunk dark yellow. Anterior
row of tubercles more prominent than the others, sometimes so long that they hang
down over the next two rows. Third segment of neck, close to tubercled trunk, viscous,
and with a little extraneous matter adhering. Rostrum slightly broader towards apex,
ciliated cup usually quite everted, the lamellae then standing far apart. Papillae on
discs, only once seen, like little curved thorns. Viscera difficult to see through the
dark, thickened skin. Under strong pressure stomach seen, with its walls filled with
small clear globules. Brain large. Glands, intestine, cloaca, and vibratile tags not seen.
Arrangement of teeth unusual. Three large teeth in one jaw fit into the spaces between
four large teeth in the other. There is an additional thinner tooth at each end of the
row of three.
In the contracted state the tubercled trunk is so similar to that of Philodina mac
rostyla, variety tuberculata, that it might be passed over for that species. When it
extends itself it is found to differ in everything else. Every part of the Philodine is
long and slender, of the Callidine short and broad, except the antenna. This is straight,
not elbowed as in P. macrostyla. The foot appears to end in a disc, as in the
' symbiotic ' Colliding. It is a very small and obscure disc, and no perforations could
be seen. It may yet be found to have short, broad toes. The tubercles could be
removed by rolling the animal under the coverslip.
Habits. — Very slow and cautious. After being disturbed it may remain fully
contracted and motionless for a long time. It puts out its head very gradually, feel
ing carefully about with its long antenna before venturing out. When it has gained
confidence it walks forward rather briskly for an animal so heavily armoured. The
very short foot is only momentarily seen, the disc not at all, unless it happens to be
walking upside down. It was not eager to feed, and when it tried to do so was
evidently annoyed by the debris surrounding it, and soon desisted.
Habitat. — In the sediment of one or two ponds at Fort Augustus, February 1904.
It was pretty abundant for a time in one pond.
Philodina laticornis, n. sp. (Plate VII. figs. 12a to 12c.)
Specific characters. — Very large. Foot and rump together about f of greatest
length when creeping. Corona narrower than trunk, discs with small central papilla?.
Rostrum short, broad, with very small ciliate lamellae. Antenna stout, length equal
to f diameter of neck. Brain fairly large, with pair of large, oblique, yellowishred
eyes. Two teeth in each jaw. Foot of three joints. Spurs large, broad, divergent,
interstice slightly exceeding diameter of spur at base. Dorsal toes small, ventral long,
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 38 1
incurved. Reproduction viviparous. Swims free, with spurs brought close together
and discs also approximated.
General description. — Greatest length ^ to ^ inch when creeping. All colourless
except the alimentary canal. Longitudinal folds of trunk few, dorsal faint, lateral
stronger. Brain elongate, triangular. Stomach ample. Intestine elongate, elliptical.
Rump and foot of about equal length. First footjoint long, with faint annular striae.
Second joint with stronger striae, crossed above the spurs by two oblique folds of
skin, which nearly meet in the middle line, and give the appearance of an extra joint.
Spurs with obscure shoulder on inner side at base, then slightly contracted and ex
panded again nearer tip. Footglands rather small, with very long ducts. Most
examples with two wellgrown young, showing teeth and corona, and one younger
foetus. Vibratile tags, five on each side seen.
Habits. — The large size, lanky form, and large spurs and toes, suggest that the
animal is a parasite, but it has not yet been found attached to any host. On the other
hand, its readiness to swim, and its characteristic attitude when swimming, spurs and
discs being brought together as though to lessen the resistance, are like the actions of
a freeliving animal. Several species of parasitic Bdelloids have small brains and are
blind, and there is some reason to believe that this reduction is a consequence of the
mode of life. The power of swimming might be of advantage to an ectoparasite by
enabling it to change its host if necessary. When swimming, the rostrum is kept fully
extended. When creeping, the toes are often kept out during the whole of the step.
Habitat. — Among aquatic mosses growing at the margin of Loch Ness, at Fort
Augustus, April 1904 ; in the Caledonian Canal, Fort Augustus, December 1904.
Philodina laticeps, n. sp. (Plate VII. figs. 11a to 11/i.)
Specific characters. — Very large, elongate, yellowish. Corona very large, much
wider than trunk, discs broad, concave, separated by space nearly equal to diameter of
disc. On each disc an elevated conical papilla, with broad apex bearing several short
motile cilia. Rostrum short and broad, with minute lamellae. Antenna short, length
equal to ^ diameter of neck. Brain a minute triangle, no eyes. Teeth, two on each
jaw, with one thinner tooth. Foot and rump together just under half of total length.
Foot of four joints. Spurs large, broad, bladeshaped, divergent, interstice equal to
diameter of spur. Dorsal toes small, ventral long, incurved. Parasitic on insect larvae.
Oviparous.
General description. — Greatest length ^ to ^ inch when creeping. All hyaline
except alimentary canal. Trunk longitudinally plicate, central segments covered with
a hairlike growth, which is probably a vegetable parasite. Corona broadest and discs
largest known in the order. Yolkmass with eight or nine nuclei ; the large egg
pointed at anterior end. Intestine long, elliptical. Footglands long, with very long
ducts. Four vibratile tags on each side seen.
382 MR JAMES MURRAY ON A NEW FAMILY
Habits. — Parasitic on insect larvae which live in running water. It has been found
on larvse of several species, adhering to the thorax, between the bases of the legs. When
separated from its host it is little disturbed, immediately begins to creep actively about,
and readily feeds. When feeding it is very restless, and sweeps the great corona from
side to side and all over the field. The apparent breadth of the corona is often in
creased by a peculiar habit the animal has of pulling in the sides of the trunk till it
resembles a stem supporting a large flower. It is then more like one of the large
headed Ehizota, such as (Ecistes velatus, than a typical Philodine (fig. llh). When
feeding it draws the rostrum in till it is depressed below the surrounding surface of the
head.
Comparison o/"P. laticeps with P. laticornis. — The two species resemble one another
very closely in some characters, and differ greatly in others. The agreement is so close
that it is difficult to avoid the conclusion that they are related animals. On this
supposition an interesting comparison of the differences of structure in relation to the
different modes of life may be made. P. laticeps is a parasite ; P. laticornis has only
been found free. They agree in general form, in the rostrum, spurs, and long curved
ventral toes, so closely that but for the longer foot of P. laticeps the same drawing of
the extended animal could represent both. P. laticornis has a large brain and eyes,
small papillae on discs, larger antenna, and shorter threejointed foot. P. laticeps has
much larger corona, very large papillae on the discs, shorter antenna, longer fourjointed
foot, much smaller brain, and no eyes. If the parasite P. laticeps has been derived from
the freeliving P. laticornis, it is interesting to note that while it has gained a larger
mouth, it has lost its eyes and most of its brain. Should P. laticornis, as is possible,
prove to be also parasitic, the force of the comparison is diminished, but not altogether lost.
The habit of swimming might enable a parasite to change its host when necessary, and
so render it less dependent, and the retention of the large brain and eyes may be
attributed to this habit.
P. laticeps is oviparous, P. laticornis viviparous. This is the only instance known
to me of closely related Bdelloids differing in the mode of reproduction.
Habitat. — In a little stream entering Inchnacardoch Bay, Loch Ness. Very
abundant during the winter of 19034. Any handful of Fontinalis taken from this
stream and shaken in water yielded thousands of examples. Early in the summer of
1904 the stream dried up, and remained in this condition till October. When the water
returned to the channel insects and rotifers had disappeared, and up till the end of
November neither had again been found. Similar streams in the same district were
searched, but though larvae were found, there were no rotifers upon them.
Philodina humerosa, n. sp. (Plate IV. figs. 4a to 4 a.)
Specific characters. — Small, dull grey, strongly plicate on trunk. Ventral trans
verse folds, fourteen or fifteen. Central setae on discs spring from large conical
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 383
prominences. Space between discs equal to half diameter of disc. Rostrum short,
shaped like an acorn, basal joint papillose. At back of oral segment, on each side of
rostrum, a large rounded papillose prominence. Length of antenna equal to diameter
of neck. Neck with rounded prominence at each side of antenna. Foot of three joints,
stippled. Spurs small, tapering, divergent. Dorsal toes tapering; ventral larger, obtuse.
Two teeth on each jaw. Oviparous.
General description. — Greatest length when creeping, y^ to t ^q inch. Skin of
trunk dull yellowishgray, opaque, finely stippled, foot more strongly stippled. The
prominence from which the central seta arises occupies nearly the whole of the upper
surface of the disc. The great papillose bosses on the back of the oral segment are
unique. They are conspicuous when the animal is creeping as well as when feeding.
The skin of the first footjoint is thickened dorsally, but does not form a boss.
Related species. — This species is closely related to Philodina alpium (Callidina
alpium, Ehr.) and P. brycei, (C. brycei, Weber). The three species form a very natural
group. They are semiloricated. The skin of the trunk is thick. Its anterior edge is
•cut into definite forms and bears six knobs or processes. Its ventral surface is crossed
by deep transverse folds, 9 to 15 in number. Though not quite rigid, it alters little in
shape. When the animal is fully retracted the deep longitudinal folds allow the
anterior edge of the trunk to be closed. In P. alpium and P. brycei the two anterior
dorsal processes of the trunk form a fork which receives the antenna, as in Anurwa and
Brachionus. In all three species the central setae rise from large conical processes.
There are four toes.
Habits. — Like its relatives P. alpium and P. brycei, it is very slow in its move
ments. When it has been left undisturbed for a time it feeds with confidence. It
ceases feeding at short intervals, but resumes again at the same spot.
Habitat. — Found in ground moss and Fridlania growing on stones. Old pier at the
Monastery, Fort Augustus, 7th February 1904. At several spots near Fort Augustus.
Not yet seen anywhere else.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 15). 56
384
MR JAMES MURRAY ON A NEW FAMILY
EXPLANATION OF PLATES.
The drawings of the complete animals are all made to a uniform scale, with the exception of Microdina,.
which is drawn larger. The separate details are drawn of any convenient size. In the descriptions the only
measurement given is the greatest length of the animal. All other measurements obtained are put into the
drawings. Whenever possible the width of the corona, collar, neck, trunk, rump, and foot are measured,
also the length of head, neck, trunk, foot, spurs, and jaws. These sizes, expressed in figures, convey but a
vague impression of the appearance of an animal unless accompanied by a drawing. They are therefore
omitted from the text.
The form of the upper lip is carefully drawn, but is not included in the descriptions, as no common names
for its various parts have yet been agreed upon.
Plate I.
1. Microdina paradoxa, n. sp.
a, dorsal view, example from L. Vennachar, 1902.
b, lateral view, another L. Vennachar example.
c, ventral view, variety from L. Treig, 1904.
d, deposition of egg.
e, /, antenna in different degrees of extension.
g, foot, showing toes and glands under pressure.
h, toes, dorsal view.
i, rostrum, ventral side.
Plate II.
a, head of Callidina papillosa, ventral side.
b, head of Microdina paradoxa, „ „
c, head of Adineta barbata, ,, „
d, jaws of Philodina brycei.
e, jaws of Microdina, form with three loops.
/, jaws of Melicerfa.
g, h, i, j, four varieties of spurs of Microdina.
k, I, m, n, o, jaws of Microdina, five views of same
pair.
k, direct ventral.
I, oblique ventral.
m, direct dorsal.
n, dorsal, under pressure, rami turned on side.
o, lateral.
Plate III.
2. Callidina angusticollis, n. sp.
2a, animal in case, feeding, dorsal.
26, side of head, feeding.
2c, jaws.
2d, head seen from above.
2e, section of oral segment at thickenings.
2/, section of neck.
2g, spurs.
2h, side of foot and rump.
2i, animal in case, in characteristic attitude.
2j, side of rostrum.
2k, front of rostrum.
3. Callidina annulata, n. sp.
3a, animal in Frullania cup, feeding, dorsal.
3b, side of head.
3c, front of rostrum.
3d, jaws.
3e, antenna.
3/, spurs.
AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 385
Plate IV.
4a, dorsal view, feeding.
46, ventral view, creeping.
4c, back of head.
id, side of rostrum.
4. Philodina humerosa, n. sp.
4e, section of neck.
4/, toes.
4g, jaws.
5a, dorsal view, showing stippling.
56, dorsal view, feeding.
5c, antenna.
5. Callidina pulchra, n. sp.
5d, side of foot.
5e, spurs.
5/, jaws.
6. Callidina crenata, n. sp.
6a, dorsal view, creeping, showing papillae.
66, ventral view, creeping, showing internal
structure.
6c, section of neck.
§d, jaws.
Plate V.
7a, dorsal view, creeping.
lb, ventral view, feeding.
7c, side of rostrum.
Id, section of neck.
Callidina murieata, n. sp.
7e, side of antenna.
If, jaws.
Ig, side of foot.
7h, spurs and toes.
8a, dorsal view, creeping
86, side of rostrum.
8c, front of rostrum.
8^, jaws.
8. Callidina crucicomis, n. sp.
8e, spurs crossed.
8/, side of foot.
8g, spurs and toes.
Plate VI.
9a, dorsal view, feeding.
96, front of rostrum.
9c, jaw.
9. Callidina incrassata, n. sp.
9d, side of antenna.
9e, spurs and disc.
9/, section of neck.
10a, dorsal view, feeding.
106, ventral view, creeping
10c, side of head.
lOd, jaw.
10. Callidina armata, n. sp.
lOe, section of head, showing toothlike processes.
10/, dorsal view of foot, showing fold and glands.
\0g, side of foot.
106, spurs and disc.
386 NEW FAMILY AND SPECIES OF ROTIFFRA OF THE ORDER BDELLOIDA.
Plate VII.
11a, dorsal view, feeding.
116, front of antenna,
lie, side of antenna.
\ld, papilla on disc.
11. Philodina laticeps, n, sp.
lie, jaw.
1 1/, spurs and toes.
11 a, back of rostrum.
11/i, characteristic attitude, with trunk narrowed.
12a, dorsal view, swimming.
126, dorsal view, creeping, showing viscera.
12. Pliilodina laticornis, n. sp.
12c, side of antenna.
Murray:
Trans. Roy Soc. Edm r Vol. XLI.
A New Family and Twelve New Species of Bdelloida.— Plate I.
M'F»cU»t 4. Ermine. L.tlEdm
MlCRODINA PARADOXA
n. sp.
Trans. Roy Soc. Ediif, Vol.XLI.
fuRRAY: A New Family and Twelve New Species of Bdelloida.— Plate II.
Trans. Roy Soc. Ediif, VbLXLI.
Murray: A New Family and Twelve New Species of Bdelloida— Plate III.
MfFarlane &. Ersfcme.Litl. Edm'
2, CALLIDINA ANGUSTICOLLIS.n
sp.
3, CALLIDINA ANNULATA.n.
sp.
Trans. Roy Soc. Edm r Vol.XLI.
Murray: A New Famiiy and Twelve New Species of Bdelloida.— Plate IV
4.
M'FaxlaTie &. Er shine. Uth.ZAm*
HILODINA HUMEROSA.n.sp. 5,CaLLIDINA PULCHRA,r,sp. 6, CALLIDINA, CRENATA.u.sp.
Trans. Roy Soc. E diif. Vol. XL!
Murray: A New Family and Twelve New Species of Bdelloida.— Plate V.
7 CalLLDINA MURICATA.n.sp. 8, CaLLIDINA CRUCICORNIS, n.sp
MrFa.rla.ne & Erskine, EiLkEdit 1
Trans. Roy Soc. Edirf, VoLXLI.
Murray.: A New Family and Twelve New Species of Bdelloida.— Plate VI.
9, CALLIDINA INCRASSATA.n.
sp.
10, CALLIDINA ARM ATA n.
K!F»lui( UrAine.IitTi.Edu
sp.
Trans. Roy. Soc. Edirf, Vol. XLI.
Murray.: A New Family and Twelve New Species of Bdelloida.— Plate VII.
11, PHILODINA LATICEPS.n.sp. 12, PhILODINA LATICORNIS, n.
KFaTlans WSrskme litt Edin 1
sp.
'
( 387 )
XVI. — The Eliminant of a Set of General Ternary Quadrics. — (Part III.)
By Thomas Muir, LL.D.
(MS. received December 12, 1904. Read! January 23, 1905. Issued separately April 15, 1905.)
(41) The in variance of the equations
a Y i: 2 + b^j 2 + c x & + f x yz + y x zx + \xy = , \
a 2 x 2 + b 2 y 2 + c 2 z 2 + f 2 yz + g. 2 zx + h. 2 xy = , V
a B x 2 + b 3 y 2 + c 3 z 2 + f s yz + </ 3 zx + h 3 xy = , J
with regard to the group of cyclical substitutions,
and the consequent invariance of the eliminant with regard to the reduced group con
sisting of the last two substitutions, has been already referred to. When the eliminant
is expressed in terms of the threeline determinants formable from the array of
coefficients, the invariance in question is selfevident, as each of the twentyeight parts
composing the expression is invariant by itself. For convenience this form may be
repeated from § 31 with a slightly improved notation. It is
0U00
+ 2l'4'8'9
+ 20077'
+ 24457'
220016
+ 21'55S'
+ 207'8'9'
21688'
220147'
 244'99'
+ 0456
21T88'
+ '20159
21'6'88
+ 0123
+ 21137
+ 20125'
+ 21157'
+ 204'5'6'
 24468
+ 0'7'8'9'
21'448
 0'456
+ 21166
+ 24o'7'9'
21489
+ 213'7'S'
2H4'9
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 16). 57
388 DR THOMAS MUIR ON THE
where
are used for
1 2 3
4 5 6
7 8 9
1 «A C 3 1;
1
1 a A/ s \,
1 Vi^S li
1  c 1 a 2 h 3
1 <hh9e li
 ftjCg/tg  ,
1 c i a 2/s 1
! «i V'a 1 >
1 V2/3 1 !
1 ^ft 
1' 2' 3'
4' 5' 6'
7' 8' 9'
1 UiA 1.
\ c \9Jh l>
1 V's/s 1 '
1 & l/ 2 .?3 1 .
1 c A/a 1 »
1 «i/2#a 1 >
I^A li
K/203 l»
1 a i9i^% 1 >
IV^/s ll
respectively.
(42) There is, however, a second form of invariance which it is convenient now to
consider. Looking at the equations we at once see that the performance of the
interchange
Tx a f\
\ y b g )
leaves them unaltered, and that the same is true of either of the interchanges
71* c
\>x a
y b g\ ~pz c h
z c h
From this it follows that the eliminant is invariant to each one of the three interchanges
ia f\ pb g\ 7K
b g J , \ c h J , ba f
Taking the first of these and observing its effect on our twenty determinants of
the third order we find that it is equivalent to the substitution
0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 0', 1', 2', 3', 4', 5', 6', 7', 8', 9'
0, 4,6,5, 1,3,2, 7,9,8; 0',  4',  6',  5',  1', 3', 2',  7',  9',  8'
In an expression, like the special form of eliminant given in the preceding §, where
each term is the product of four of these determinants, and where, therefore, the sign
of each determinant may be changed with impunity, this substitution has the same
effect as the simpler substitution
1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 1', 2', 3', 4', 5', 6', 8', 9'
4 , 6 , 5 , 1 , 3 , 2 , 9 , 8 , 4', 6', 5', 1', 3', 2', 9', 8'
or the interchange
Similarly we have
1 , 2 , 3 , 8 , 1', 2' 3', 8'
4 , 6 , 5 , 9 , 4', 5' 6', 9'
Tbff\ ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 0', 1', 2', 3', 4', 5', 6', 7', 8', ?
,  6 ,  5 ,  4 ,  3 ,  2 ,  1 ,  9 ,  8 ,  7 ;  0',  6',  5',  4',  3',  2',  1',  9',  8',  7'
c:oc
ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 389
and therefore in the case of products of four
1,2,3,4,5,6,7,9; 1', 2', 3', 4', 5', 6', 7', 9'
6, 5, 4, 3, 2, 1, 9, 7; 6', 5', 4', 3', 2', 1', 9', 7'
a, 2,3, 7; 1', 2', 3', 7'\
6,5,4,9; 6', 5', 4', 97 '
and, lastly, in the same circumstances
c h\ \fl, 2,3,7; 1', 2', 3', 7'
a f) " \5, 4, 6, 8; 5', 4', 6', 8'
(43) A comparison of the three interchanges, which, in the case of a fourfactor
product, we have thus found to be equivalent to
Ta A fh g\ te h\
\,b gj, \c h), \,a f)
respectively, leads at once to the further observation that if the expression in which
the interchanges have to be made be invariant to the cyclical substitution, the three
interchanges are not essentially different. So far, therefore, as the above eliminant is
concerned, we need only consider one of the interchanges, say the interchange
/1,2,3, 8; 1', 2',3',8'N
U , 6 , 5 , 9 ; 4', 6', 5', 9' ) ,
it being borne in mind that this implies that the determinants
0,7; 0', 7'
are invariant to the interchange. The determinants 0, 0', which are invariant to the
cyclical substitution as well, we shall therefore speak of as being doublyinvariant.
(44) Turning then to the eliminant and applying this interchange to each of its
twentyeight parts, we find that twelve of them, viz., the
1st, 2nd, 3rd, 4th, 5th, 7th, 11th, loth, 18th, 21st, 26th, 27th
are doublyinvariant ; that twelve others may be grouped as six binomials which are
doublyinvariant, either term of each binomial being produced from the other term, viz.
6th and 8th,
13th and 14th,
16th and 22nd,
17th and 23rd,
19th and 20th,
25th and 28th ;
and that the four remaining parts (the 9th, 10th, 12th, 24th) are
20125', 04'5'6', 0'456, 24468.
Now we can show (see §39) that 04'5'6' — 0'456 is expressible as the difference of two
terms which are each doublyinvariant, viz., the difference 07'8'9'  0789. Further, since
20125' = 21129 + 21246
390
DR THOMAS MUIR ON THE
the other two terms are expressible in the form
(21129 + 24468) + 21246,
where the binomial and the single term which follows it are both doubly invariant.
There is thus finally obtained an expression for the eliminant which shows its property
of doubleinvariance, the constituent parts being fourteen single terms and seven
binomials, viz.
0000
+ 0123
+ 0456
220016
21129
 24468
+ 20077'
+ 21137'
+ 24457'
220147'
2H4'9
2T448'
+ 220159
+ 2U5'7'
+ 21'558'
+ 4 07'8'9'
211'88
 244'99'
0789
+ 2137'8'
+ 2457'9'
+ 0'7'8'9'
+ 21166
 21489
+ 2l'4'8'9'
 21688'
 21'6'88'
(45) Let us return now to the eliminant of the 10th order as obtained in § 9, viz.
«i A
ffi h i A
9i A «i
h \ ih
• 9i h A
a 2
A
C 2
h.
'.12
\
0i
A
CLc,
•
h. 2
h
G S
9 S
^3
A
«3
A
C 3
•
h s
9 3
h
fh
A
«3
h
6 4' 5' 6' + 0'
By transposition of rows and of columns, and by altering its sign, this is readily
changeable into
6'
1 c l
*i
9i
•
A
2 '*2
h
92
A
3 C 3
*■
9z
A
1
°i
h
c i
A
9i
2
a 2
h
C 2
A
r/ 2
3
«3
h
C 3
A
9z
9j
A
a i
h
c i
h
9 2
A
a 2
h
C 2
K
■ 9 3
A
«3
\
C 3
h.
' 6
4
.
4'
5'
5
+ 0'
ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS.
391
— a form more suitable for obtaining the cofactors of the elements of the last row in
terms of the familiar threeline determinants 0, I, 2, ... . Taking first the cofactor
of + 0', and using Laplace's expansiontheorem, we find it
= 0(0016)7(89') + 4(02 17') 3(05 + 69') + 9(78)  8'(25  7'9'),
= 000 2016 2147' + 2789 + 7'8'9'.
Similarly the cofactor of 4 is found to be
002 + 038 + 056 + 03'6  189 l'48 + 224
 255' + 3'47'  4'99'  559 + 5'7'9' + 669' + 6'89,
and the cofactor of 4' to be
004+ 013 +036'+ 067 119 11'4+1'8'9'
+ 244 + 26'8'  25'7  2'99' + 339' + 3'79  579 .
The full eliminant is thus
(0 + 0')[000  201 6  2147' + 2789 + 7'8'9']
002 + 038 + 03'6 + 056  1'48  189 + 224
255' + 3'47'4'99'559
+ 2*'
004+ 013 +036'+ 067  119  11'4 + 1'8'9
 1'48 189 + 224
' + 5'7'9' + 669' + 6'89j
+ 244 + 26'8'  257  2'99' + 339' + 3'79  579
This does not, of course, differ from the form used in § 41. As a matter of fact, it will
be found on examination that nineteen of its thirtyeight terms agree with terms in the
expression of § 41, and that the other nineteen can be changed without much difficulty
so as to establish the identity of the two expressions.
(46) As may be supposed, however, the importance of the new result does not
consist in its affording a verification of that previously obtained. It is more interesting,
in fact, in its unsimplified state ; for it has now to be noted that each of the three
lengthy expressions found in it as the cofactors of + 0', 4, 4' can be put in the form of
a simple threeline determinant. For example,
58'
+
67
5
+
69'
48
6
59
4
9
A 47 '
0+ ir
+
58
o«
+ *1 . « . 5 1 + 789
5 6 4
±«)t.
O^' + f + £) + 0(^ + 5 "' + «T) ; , ,,,
,59
Z A »u» _ V* ^699'
= 000 + ^05
08'  79
i>f
08'  79
4 J ' ^"\ 4
= 00020532369' + 7'8'9' + 2789 ,
= 00020162147' + 7'8'9' + 2789,
= cofactor of 4 0'.
+ 7'8'9' + 2789,
392
DR THOMAS MUIR ON THE
Similarly it may be shown that
+
8
59
4
2 55 '
4
►?
9
o 66'
3 " 5
48
'6
+ 47 '
(i
6
and
=  002 + 038 + 03'6 + 056  1'48  189 + 224  255' + 3'47'  4'99'  559 + 5'7'9' + 669' + 6'89
= cofactor of 4 ;
+
28'
27
22'
5 "T
h 3
3
9
o + ™
6f
19
2
7
*T
= 004 + 013+ . . . .
= cofactor of 4'.
It consequently follows that the eliminant may be put in the form
(0 + 0')
+
58'
8
59
4
O
2*
8
59
4
55'
2*'
<
27
3
•7
67
5
<
9
+ f
5
9
3?
9
n 39 '
0+ 1
<t 33 '
6 f
7
48
6
<
48
"6
47'
+ lT
T
19
2
7
*"T
or, say, for convenience of future reference,
(O + O^Aj  ±4A 2  ±4'A 3 .
(47) A glance at the cofactor of + 0' as first obtained, viz., 000 — ^016 — . . . .
shows that it is invariant to the interchange of § 42. Since, therefore, the performance
of this interchange on the determinant form of it, A 1? — where the in variance is not in
evidence, — cannot make any alteration in substance, we shall obtain thereby a second
determinant form V l5 thus arriving at the unexpected identity
+
58'
59
"4
67
5
+
48
6
69'
+
47'
04 9 '
9
38
1
=
27
3
A 28 '
8
7
19
2
o + ;
Further, since the whole eliminant is invariant to the interchange, the removal from
it of the invariant portion (0 + 0') A T must leave a portion which in substance, if not
in form, is likewise invariant. This consideration gives us the identity
ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS.
393
or, say,
2 4
2i
8
59
4
55'
2 ~T
+ 2 4 '
•Tt
27
3
3
+ 6 /
5
9
o 66'
6 ~~5
9
+ «T
6f
48
6
<
>4 4 '
19
2
7
*¥
9
38
1
. 33'
6 T
o
+ 2»'
5
67
5
3f
n 28 '
+ T
8
22'
5 ~T
8
»4 8 '
2 55 '
4
19
n *?'
„ n '
48
i 44 '
2
+ ¥
4 "T
6 "
7
X ~T
±4A., + ±4'A, = £lV„ + ±1'V„.
If we denote the first portion of the eliminant, viz. (0 + 0') A : by A, and the second
portion by  B, and the alternative forms of these by A' and  B', we can thus express
the eliminant in four different ways, viz.
AB, AB, AB', AB'.
(48) Using the cyclical substitution on the righthand member of the immediately
preceding identity, we see that we can put in place of it
2>
7
19
2
*"T
+
22'
<
48
"6"
f
39'
i —
' 1
9
. 33'
6 ~ 1
9
A 69 '
0+ 5
o 66
6 ~ 5
27
<
22'
59
8
2™
4
3
5 ~3
4
But the determinants here are those occurring in the lefthand member : consequently
we deduce
±(42'A 2 = ±(24')A 3 .
(49) Returning to § 46, and noting that A 2 has two columns in common with A ]s
and that the result of the cyclical substitution on A 2 is simply to change this pair of
columns into another pair of A l5 the third column remaining all the while unaltered, we
see that
(0 + 0')A 1  24A 2
394
DR
THOMAS
MUIR
ON THI
can be put in the form
a 58'
+ T
8
59
4
2* 5 '
4
67
5
0+£
5
9
„ 66'
1~1
7
48
6
<
ii 4 '
6
4
5
6
+ 0'
A new expression for the eliminant is thus
<
7
48'
6
2*'
a 2 8'
+ T
27
3
B .y
59
4
4
8
4
9
o + *
6 "T
9
67'
5
5
66'
d 5
19
2
7
4L 1 '
2
6
4
5
+ 0'
This can be further improved by removing from the fourline determinant the
terms containing 0', and associating them in the form 0' V x with — 24'A 3 just as 0\ has
been associated with — 24A 2> the final result being
+
47'
7
48
6
¥'
°4 r
19
2
59
4
<
8
2 A 5 '
4
+
8
f
9
67
5
<
o 66'
3 "¥
38
1
9
6
4
5
4'
5'
+
27
3
39'
1
6'
4
6
11'
2
22'
3
33'
1
0'
From this, by making the now familiar interchange, we obtain the alternative form
+
17'
7
19
*¥
•4 r
48
7
i 44 '
h J
2
"6"
] fi
38
1
A 39 '
o+ r
9
.»'
+
9
69'
j —
5
67
5
o 66'
d ~ 5"
8
27
3
28'
+ 3
f
59
4
8
4
255'
4
2
1
3
1'
3'
2'
0'
Here, however, if in both determinants we transpose the second and third rows, and
thereafter the second and third columns, we find that either determinant has three rows
in common with one of the determinants of the form from which this was derived.
Subtraction is thus readily accomplished, the result being
ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS.
395
<
7
48
6
'4
59
4
°4 8 '
8
■) 55 '
" 4
9
67
5
°4
„ 66'
5
61'
42'
53'
+
17'
19
7
*4
*T
2
8
A 28<
0+ y
27
3
22'
5 y
38
1
9
. + *
a 33'
6 y
.4'
35'
16'
This is a verification of the identity
±(42')A 2 = ±(24')A 3
already obtained in § 48. The latter form of it shows that both of the fourline deter
minants are invariant to the cyclical substitution ; and as the interchange of § 42
transforms the one into the other, it follows that both are doublyinvariant.
(50) The two new forms of eliminant just reached make clear the fact that if one
of the four sets of determinants
4,5,6 ; 4', 5', 6' ; 1,2,3
1', 2', 3'
vanishes, the eliminant takes the form of a single fourline determinant. For example,
if 4, 5, 6 have each the value zero, the eliminant is
+
17'
19
^
11'
H T
2
i
2
8
°4
27
3
22'
3
38
1
9
39'
+ T
33'
2
4'
5'
6'
+ 0'
We are thus brought to consider the problem of finding the set of four equations whose
coefficients are the elements of this determinant. In the quest for a solution we are
not without a lead, since for one of the very special cases brought into notice by
Sylvester the desired set of equations has already been obtained. *
(51 ) From the fundamental set of equations there can be deduced (§ 33)
 u x c 2 h 3  = 9x 2 + 2y 2 7'yz+l'xy = 0,
j ?/ 1 a 2 & 3  = 0z 2 + lyz+4zx + 7xy = 0,
and from these by multiplication by z and y respectively we obtain two equations in
volving the desirable facients yz 2 , zx 2 , xy 2 , xyz, together with the undesirable y 2 z. On
eliminating the last mentioned there results
(02 + 17>2 2 +19ac 8 + 27*y 2 + (24  W)xyz = 0,
and by cyclical substitution
38^ 2 + (03 + 28V 2 + 27av/ 2 + (3522')x?/z = 0,
38 ? /z 2 + 19zx 2 + (01 + 39>(/ 2 + (1633>2/2 = 0.
* Proc. Roy. Soc. Edin., xx. p. 377.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 16).
58
396
From these we obtain
DR THOMAS MUIR ON THE
(o + ' 7 <y
9z.r 2 +
1 ,;
lyz* + (0 + 8')z* 2 +
7xif + Ui l'te =
9 / 2 A
y82/« 2 +
9za! 2 + (o + y 9 'W + ( 6 ~ y 3 ') x ^ = °
The performance of the interchange of § 42 gives the companion set
8fz +
\lxhj + (0+^9'V* +
+ J 8' )a
6„ „ . /„ . 6„A«
9z 2 .c + ^2^5'W =
9z 2 z + (3  1 6'W =
7x 2 y + ^8yh + (o+±7'\*z + (l  ^4'W
= 0.
The necessary fourth equation for the cases alluded to in the preceding paragraph
is got from the cubic M 3 of § 9,
ix 2 y + byh + 6z 2 x + Myz 1 + b'zx 2 + <o'xif + (0 + 0')xyz = ,
or from its companion M' 3 ,
•2'x 2 y + 3'fz+Vz' i x+2yz 2 + 3zx 2 + lxy 2 + (0 + 0')xyz = 0,
by putting three appropriate coefficients equal to zero.
(52) Another special case of similar type is still more interesting, viz., the case where
7', 8', 9' vanish. The Jacobian of the given set of equations, viz.
2(8'x 3 ) + 2(24 + 2> 2 2/ + 2(22 + i')yz 2 + (40 + Q')xyz = 0,
then loses three of its terms ; and as the operation 2M 3 + M' 3 gives
2(24 + 2> 2 // + 2(24' + 2)yz 2 + 3(0 + 0')xyz = 0,
it is clear that there follows
2(24)v/2 2 + (020>?/z = 0,
— an equation which can be used to complete the first set of three in § 51. Since the
vanishing of 7', 8', 9' makes 11', 22', 33' identical, the resulting eliminant is
19
2
?? 9
7
27
3
42'
53'
61'
24' 35' 16' 020'
EL1MINANT OF A SET OF GENERAL TERNARY QUADRICS. 397
Had we used the operation M 3 + 2 M' 3 we should have obtained the alternative form
8
59
4
24'
67
5
9
35'
7
48
6
1 6'
42' 53' 61' 020'
In this case, however, the two forms are not even superficially different, the one being
obtainable from the other bv changing rows into columns and attending to the
identities
26 = 89 , 34 = 97 , 15 = 78:
or,
159 = 267 = 348 = 789.
( 399 )
XVII. — Theorems relating to a Generalization of Bessel's Function. II.
By the Rev. F. H. Jackson, B.N. Communicated by Dr W. Peddie.
(MS. received February 6, 1905. Read February 20, 1905. Issued separately April 18, 1905.)
§ 1. Introduction ....
§ 2. Function ~E p (x) ....
§ 3. Expressions for Jacobi's Functions
CONTENTS.
PAGE
399
§ 4. J'„» . .
400
§ 5. Various series
403
PAGE
405
407
1.
Introduction.
The theory of the functions commonly known as q functions might perhaps be
greatly developed, if investigators were to work on lines suggested by the functional
notation of wellknown analytic functions. For instance, the analysis connected with
the circular functions sin x , cos x , . . . , might be regarded as the theory of certain
infinite products without using any special functional notation. It need not be
explained however, how great was the gain to elementary algebra by the introduction
of the exponential function (regarded as the limit of a certain infinite product, or as
the limit of a certain infinite series) denoted e x , with certain characteristic properties,
enabling the worker to make transformations easily and quickly. Of course, the vast
store of interesting and in many cases useful results connected with the elementary
functions of analysis might have been obtained without the introduction of any
•notation capable of rapid and easy transformations, but I think it unlikely that they
would have been obtained.
In chapter xi. of Cayley's Elliptic Functions the identity
1g 2 !^
. 1
f" ) 1  q* ' 1 _ r/ 2«+2 T I _ q 2,\ _ q i ■ 1 _ (? 2..+2. 1 _ q 1n+i
+
^,<rip> 0:
is used in order to express Jacobi's function, in the wellknown form
1  2q cos 2x + 2<? 4 cos 4x  2q 9 cos Qx + ....
The likeness of the series ( I ) to Bessel's series is very obvious. It is a very special
case of the series which I have denoted J [n] in previous papers, and in itself might have
suggested a theory of q functions analogous to Bessel's functions. In the discussion
TRANS. ROY. SOC. EDIN , VOL. XLI. PART II. (NO. 17).
59
400 THE REV. F. H. JACKSON ON
of q functions a great variety of notations has been used. I propose in this paper to
bring before the Society a series of formulae relating firstly to a function E i ,(x) analogous
to exp (x). These formulae are supplementary to those given in Trans. Roy. Soc.
Edin., vol. xli. pp. 105118, and will lead to one or two interesting properties of a
function J p m , n (x), which may be termed a generalized Besselfunction of double order,
and to various novel expressions of elliptic functions in terms of the generalized Bessel
function. For example, Jacobi's O function is expressed by the form
• (A)
in which
*(
2K«\
7T /
_ q I[»]( M )I[»
2o
= fi (1  q~ m )
m=l
W =
"Fl
V =
= » Jv P i*
FT
n =
K'
It is noteworthy that n (the order of the J functions) is in (A) an arbitrary number.
It appears only in the expression on the right side of that equation. A definite integral
expression for the functions J will also be given.
2.
Function E p (x) .
The series
and its equivalent product
1+— +p +
P — 1 p  lp 2  \
(lx)(lpz)(lp*x)
are well known : we derive a function analogous to the exponential function.
(Cf. Trans. R.S.E., xli. p. 116; and Proc. L.M.S., series 2, vol. ii. p. 194.)
iT'W kr
■E 1 (x) = l+£+p r ^ r + .... +.p*vpz +
X . X* , X
[r] = (p r l)f(pl)
The function E p (x) may be regarded, like the exponential function, either as the
limit of a certain infinite series or a certain infinite product. The results numbered
(2) . . . (26) are either easily obtained or are known in other forms.
THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 401
E»E L (x) = l (2)
p
E»E,( *)  I + ^ (1  P y> +T ^L— t (1 ?)< + (3)
"*(".£l) w
which reduce, when p= 1, to exp (x) x exp (x)=l,
E.(*)E.(J!LJ) = l + * a+ ^ t 7 + (5)
p \piJ "V pv ip 2 \p 2 lp*
,.2
= 5(1 a^p"") • • ■ • • ( 7 )
m=l
The product is absolutely convergent if  p  > 1. The series are convergent, however,
P
if  p ^1, and also for  /> j < 1 provided x <^
It follows that
if \p\ > 1, «, = (!)
E^^E^  o^E^oAe^ _ ^ E i) (co^)E J) (o>"x) = E ;)2 „^» ( ^ r l^ n ) . . (8)
The corresponding theorem in case p <1 is easily obtained by inversion of the base_p.
= n [\xY m \ 0°)
m=l ( )
Ei/^^y E L f  ^S) = n 1 1 ! rv m_1 !• • • • • • (U)
p\»l/ v \ P—IJ m=l I J
= ? i> vl)(p*l)...(p a "l) • • ■ (13)
• ■ ... (H)
n^J • E ^f3p2 J = ^ (1 aM)  i  2 p(* 2 + *~ 2 ) + 'M* 4 + *')  2PV + *~ 6 ) +
m=l
On putting x = e™, the series on the right becomes Jacobi's function
e( 2K "
in which ji4^
J9 = e K
E^fr)  1 + <!!!+* + (' + gj^^ + ■ • • • (15)
»<1, .t< , y unrestricted
lp
(Cf. Proc. Edm. Math. Soc, vol. xxh. (8).) or p>l, y< , x unrestricted.
pl
402 THE REV. F. H. JACKSON ON
From this we derive
^h(A) 1+w ' +M PTr ] ^ + <16)
/>> <i7)
*GqMC£i) I+ ' + " + " • • <18 >
• V' j \p ■
l
lx
(x<l) . . (19)
In this expression we notice that inversion of the base p simply interchanges the E
functions in the product on the left side of the equation (18).
E ^H~)  1  [»> + 1^*  P lnIn ~$ l  21 * + (20)
= #("*> (21)
Hence
The equations
are special cases of (25).
If n be infinite
f\x) = f (v n 'A (22)
(z)x/ (a:)/ (x 2 ) (23)
^IW^K^^ 2 ) ... . (24)
_/%)x<K(*) = l (25)
e* • e  * = 1
E».E i (x) = l
/>>= e v(^t) ■'.••■.• (36)
Function I [n] (.x) .
It is well known in the theory of Bessel's function that
* t t \  x " i i , 2« + 3 2 2u + 5 3 I
€ Pl * W ~2.r(fi+l)1 !C + 2(2» + 2) a: ' 2 3(2n + 2) a! + J
I„(a;) = t n J B (» a! ).
In a paper on Basic numbers applied to Bessel's function (Proc. Loud. Math. Soc,
series 2, vol. iii., 1905), I have extended this theorem in the form
THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 403
E p (x)I m (x)
= E,(s)I [n] (z)
; " l.l x + [ 2 » + 3 ]^ 2_ [2» + 5] x5 I (27)
{2m}! 1 [2][2» + 2] [2][3][2» + 2]
![»](*) = *~"J[»](*«)
/[„](*) = *"?[„](**)
the conditions for convergence being as follows :
Case i  p  > 1 E p (x  ) and I [n] (a;) are absolutely convergent for all values of x.
Ei(«) and / [n] (#) are absolutely convergent if x< *
p p — l
Case ii  p  < 1 Ej(a;) and l[ ni (x) are absolutely convergent for all values of x.
p
E p (x) and I w (x) are absolutely convergent if x< •
\p
The series (27) is convergent for all values of p.
It is easily deduced that
l [n] (;x) = E p (ix)E p (ix)% in ix) (28)
 rn] (.T) = E J (^)E 1 (^)J [n] (x) (29)
p p
Prom these relations some interesting expressions for various elliptic functions may be
found.
3.
Relations with Elliptic Functions.
By means of equation (29) we are able to write
Iw^OIw^^) = E L (tirf)E L (  ixt)EiJixt^)}L L {  ixli) . . . (30)
"["]( a 'V"[>'](' C ^ ) P P P P
Replacing x by u, (v = ixjpjip — 1)), we obtain by means of result (11)
Iw KJImC"*" 1 ) = n ((i^y')(iAy"') I . . . (31)
Using result (12), the right side of this equation may be written
or
= E 1 ( r 1 .^)xE 1 f r V^) .... (33)
404 THE REV. F. H. JACKSON ON
This expression, when expanded in a Laurent series of ascending and descending
powers of t, takes the form (Trans. R.S.E., vol. xli. p. 117 (m)),
/ i'(^) + Z (  1 ) m ^ 2m+r2m ^ m2/ '"'(^i) • • • (34)
In case x = 1, the product II ( (1 xHY m ~ l ){l xH^p^' 1 ) )
may be expressed as
I.
n(i_p2»»)
m=l '
p (^2 + ^2) + ^4 + r 4) _ , _ . (35)
(Cf. Cayley's Elliptic Functions, p. 297, ed. 1876.)
We see incidentally that
1 1
n(ijo 2m ) l Vi
•.xWr^ ( 36 )
m— 1
for all positive integral values of n. Denoting the nature of the base by an index, we write
#\) m \p  1) m \p  1
^L4T)=^t L r) = ^(A) = (37)
I'd— W 11 ^— )=^ 4li (r^) = < 38 >
which is the expression in generalized Besselfunctionnotation of the wellknown result
1 I 2n+2 I °° 1
1  ? 2 • 1  2 * ... 1  q 2m I (1  ? 2 )(1  g 2n+2 ) I mi(l  q 2m )
On replacing t by <^ 9 , the equation (31) becomes
19 = n(l 2x 2 cos2fe 2m  1 + a;V m " 2 ) • • (39)
i9\
Using now Jacobi's notation, and writing u = *P e ix ,v = *P e' ix q = U(lq 2m )
pl pl '* m = l *
q — p, we obtain
q M^) q(MA (40)
J m( M )J[»](^) V TT ) K '
2^smx.f ] (« n )W = H /2K ? \ _
Jw(^ l «)J[»](l» J t») V 7T / V '
2gi cos x IE . SMMU&WMnUifto) _ JZKx\ u „
j¥ ^b^\¥rb^  4—) .... (44)
THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 405
We notice that in the expressions for sn, en, dn, two arbitrary constants (orders
of the functions) n, m appear :
Jacobi's function Z(x) = , / is related to the J functions as follows,
2K(^1 ) . y rlK^ = pix i » _ J» I _ prix < f » _ Jjy) 1
^ v  y ' ![»](«) Jw(«) ' < Iw(») JwW '
It is plain that Weierstrass's functions <r, £, p, may be expressed by similar formulae.
For convenience of printing, the order of the functions will sometimes be expressed
by n instead of [n]. The known formulae of Jacobi's functions will, it is evident,
give rise to corresponding forms in the case of the J functions : for example
sn 2 + en 2 = 1
gives rise to
jl(P l «)Jl(P i «) Jl(*>>*«)Jl(V J t;) WfMJHv)
(46)
= u & e" , v = — ^e ix ,p = q = e n k •
jj  1 ^  1
Using (11) and (12) it is easily found by the method of § 10, p. 1 16, vol. xli., Trans.
R.S.E., that
jl { 1  Sap*" 1 cos x + ay' J = ^(^l) + %JjT cos nxlj^^) .
By Fourier's theorem we write therefore
P^ZTt) = <T~ { n ( X ~ 2a P 2m1 cos * + aV"" 2 ) I cos nz • dx . . (47)
"Jo
In case a = 1 , this reduces to
«<^i)=iJ o cos ^ e (v> • ■ < 48 >
4.
Function J p (x) .
n,m\ I
Forming in a series, according to powers of t , the product
J M (xt) x Jp^arf" 1 )
we obtain
in which
J^(*) + 2(l) m (< am + < ,m )0*) • ( 49 )
n ' m X ~ fr '{2m + 2r* + 2r}!{2m + 2r}\{2n + 2r}!{2?'} !
{2r}! = [2][4] .... [2r]
406 THE REV. F. H. JACKSON ON
In the same way if we take
? ,( ' C) = ^ ( " 1)r {2m + 2n + 2r}!{27n + 2r}!{27 l + 2r}!{2r}! / ' 4r<m+ " +rl ' " (51)
which is related to J nm by inversion of the base p, since
J? (x) = p 2{m3+ "' +mn) p> (x)
,,,m " mX
Taking the product of two J series we find
! l „3(*0><I[»j(^ 1 ) = S^ (^) + 2( 1 )V m(m+n, (^ m + ^ i!m )I* m (a;) . . . (52)
In a previous paper (Trans. R.S.E., vol. xli. p. 106) it has been shown that
j,„ )W ) x i,„w~ ± <  n {M j^^ { \ t +2rmr) r +" ■ • p»»
There is a certain similarity of form among the series (50), (51), (53).
Consider now the product of four J functions
J M (arf) • J m (xt) ■ Mxt 1 ) ■ f^arf" 1 ) .... (54)
This expression may be written in two other forms. Firstly, by (49) and (52) we
write it
Secondly, by means of (53) we express (54) as
1 V ( _ iy {2n + 2v + 4r}! ( x t) m+ * +2r I •
I fbi / {2« + 2v+2r}![2ra + 2r}!{2v + 2r}!{2r}P ' )
If/ iy {2w + 2v + 4r}! , f _ 1)n+v+2r ) .
l£tf V ' {2n + 2r + 2v}\{2n+2r}l{2v + 2r}\{2r}\ K ' /•■• • V»)
Equating coetiicients of powers of t in (55) and (56), we find from the terms
independent of t
y( {2n+2v + ir}\ \ 2 x 2 "+'+ tr = J P P +2Y p 2mim+ ^J p P . (57)
^rf,\{2n + 2v+2r}\{2n + 2r}l{2v+2r}l{2r}\J "•"■° ^ "•»*»
The terms in the series on the left side of (57) are the squares of the terms in (53).
Generally
y {2n + 2v + ir} \{2n + 2v + 4???. + 4r} ! a ; 2 "' +2 ' ,+2 '' +4 '
^{2wi + 2n + 2v + 2r}!]27» + 2v + 27}!{2m + 2»42r}!{2ra + 2v + 2r}!{2m + 2r}!{2n + 2r}!{2v + 2r}!{2r}!
_ "V n'imim+p) j J 1 ' %" . n *mr+2r(r+n) T* 9 P I .... (58)
r=l ' '
THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 407
5.
In this section of the paper I propose to state briefly some results which may
be deduced by means of (53), (28), (29),
\L p (ix)T£ p (  ix)
r»2 /y.4
= 1  ^ ^< 1 ^ + ^^V^ 1 "" ), '"
= J m {px)JU x ) + HiW^i^) + 2 P iJ m(P x ) J m( x ) +
Indicating the nature of the base of each function by an index, we write
V ( x \ i\ p2 ( * 2 \ + 2 V Di p2 V  % v ( X
whence by (29)
■p ( x
l) J S(ynri) + 2 2 J p»o • <C  H^ri) E A " ^i)K(^ri
by (29)
by (4) and 12) ^ {p * +1)sr ^~ (1 +p)S , ^.
From (19) we find
(lxH*)(\zW)
■Jp
'H£W£i
1 cc*
1
1 ~ 2cc 2 cos 20 + ., 4 I
 I [0 j+2cos2^I m + . . . I J J m  2pcos20/ [1] + . . . 1
I ^4 2= [T] l[0]ra+ 4T] lco]l[3]+ +P rlT  1 %»&v ■ ■
E p (ix)E p (  ix) = { J„ } ' + W { J U]  2 + . . . . + W^j { J w } 2 + .
TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 17).
(59)
(60)
(61)
(62)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
60
408 THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION.
W 2 M J "! 2+ HM 2  ={Z(^ (1 _ p2)(1 .X..(iW y " ir }
{z (1 ^^^:: a (1 ?r (  1)2 1 (71)
W 2 [1]N} 2 + +^f{M 2 
= { 2( i )' (1 _^ )(1 _ i ^ . . . (i^^p 1 ) 1  1
/ V 2(l+p2)(l_ p 4 ) (1 +^0 ■ (1 +j? 2 )(l +^) . . (1 + j^) _ 1)9r I , 72 v
1 ^~ {(1pW^V^ • (i p 2r )\ 2 ^ < K J
(Cf. Trans. #.£.#, p. 110 (26).)
J[o](*)J[o](2/)  2 Mi](^)J[n(2/) + 2p 4 J [2] (a;)J [2] (y)  . . . .
""I* fT^ x > + 1^2.174^ *> "M [2]» + [2]»[4]« J ^
V^ + W t ^ ^«*J) 1 [2] ^ +[4F M +[8]J ^ + I • • (74)
X' n ~ l [4:71 + 21
E p (ix)E p (  ix ) c 2n y< 2n _ 2 y = J[»i]J M + ■ [2] J[ " ]J[ « +1]
S ( S du , <  1 [4ra + 2][4?i + 4] [4« + 2s2] 7 T n ~.
+p s(° i>[4 re + 4s  2] L JL j ^ ^ l _ JJ [m+s] J [m+s _ 1] (75)
and a similar form for x 2n . (Cf. Proc. Lond. Math. Soc, series 2, vol. iii.)
^ ( } ~FT P [^]"[2] s r^)1 ^ (  !) (i j>")(ij>*) • . . (iW p_1) f
( ^2(1 +F 8 ) • • (l+^ 2 ' 2 )(l+p) . . . (1 +/' 1 ) T ,, 1 1 (7g)
(Cf. Proc. Edin. Math. Soc, Theorem of Lommel, vol. xxii.)
It is plain that great numbers of such theorems may be found and expressed in
various forms by means of the transformations belonging to E^a:)*, but the examples
given above will suffice to illustrate the notation.
( 409 )
XVIII. — On Pennella balaenopterae : a Crustacean, parasitic on a Finner Whale,
Balasnoptera rnusculus. By Sir William Turner, K.C.B., D.C.L., F.R.S. (With
Four Plates.)
(Read February 6, MS. received February 8, 1905. Issued separately May 26, 1905.)
CONTENTS.
Introduction 409
External Characters of the Female . . 412
Chitinous Envelope 414
Structure op Head 414
Alimentary Canal 419
Nervous System . . . . • . . 422
Pennate Appendages 423
Reproductive Organs 424
The Male 427
Comparison with other Species . . . 428
conchoderma 430
Bibliography 431
Explanation of Plates 432
Introduction.
In September 1903 I received a bottle containing twelve specimens of a large
parasite presented to me by Mr Chr. Castberg, the manager of a Norwegian whaling
company which has established a fishing station at Ronasvoe in the north of Shetland.*
In his letters Mr Castberg stated that the parasites were attached to a Finner whale,
which, from its size, the mottled character of the whalebone and the pointed head, was
obviously a Razorback — Balsenoptera rnusculus. The parasites were numerous, and were
fixed to the back of the whale, and the attached end penetrated through the skin into
the blubber. Although Mr Castberg had seen many hundred whales, this is the first
occasion on which he had met with this form of parasite.
From the characters of the specimens I concluded that they were a giant species
of a parasitic Crustacean, of the family Lernseidse, and on further investigation I
associated them with the genus Pennella (Oken).
This genus is now regarded as including those members of the Lernseidse which, as
studied in the females, have the head stunted and clubshaped, with hornlike arms
radiating from its base ; the body elongated, cylindriform, not bent into a sigmoid shape ;
the anterior part of the body attenuated, but widening further back ; a pair of genital
openings with depending ova strings ; the terminal part of the body caudate, giving
origin to the characteristic bristlelike pennate appendages ; pairs of minute rudimentary
feet springing from the ventral surface of the body close to the base of the head.
From the time of Aristotle, naturalists had recognised that the Tunny and Swordfish
were infested by wormlike parasites, fastened to the skin near the fin. Rondeletius,
* I am indebted to my valued correspondent, Mr Thomas Anderson, merchant, of Hillswick, Shetland, for
putting me into communication with Mr Castberg.
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 18). 61
410 SIR WILLIAM TURNER
Gesner and Salvianus, in their respective treatises, written in the sixteenth century,
described such parasites, and Rondeletius and Gesner figured specimens from the
tunny.
Boccone published in 1674 an account of parasites found on the swordfish, Xiphias,
implanted in its flesh, which he named Sangsue or " Hirudo cauda utrinque pinnata,"
and he gave a figure. It would seem as if this animal was different from that described
by Rondeletius and Gesner. Boccone had figured a very interesting object, named
by him a " poux " or " pediculus," as big as a pea, attached to the ventral surface of the
parasite, immediately in front of the genital openings. He stated that it was fixed as
firmly to the parasite as a limpet was to a rock. I am disposed to regard this socalled
" pediculus " as the male of the female parasite to which it was attached. Its small size
compared with that of the female, and its position and attachment close to the genital
openings, corresponded with that of the male of the parasitic crustacean, Chondra
canthus lophii, described and figured by Dr H. S. Wilson and myself in 1862.
Linnaeus, in the Sy sterna Naturw, 1758, classed amongst the Vermes Zoophyta the
genus Pennatula or Sea Pens, and he named the parasite described by Boccone, which
infests Xiphias, JPennatula Jilosa. In 1759 J. L. Odhelius contributed to the Amceni
tates Academics of Linnaeus, a dissertation entitled " Chinensia Lagerstromiana," #
in which he gave the characters of Pennatula sagitta (p. 257, and fig. 13), a parasite
infesting Lophius Iristrio, the seabat of the China Sea. John Ellis reproduced in
1764 Boccone's figure of P. Jilosa and Odhelius's figure of P. sagitta. In 1802
Holten recognised a parasite on the flying fish, Exocsetus volitans, which he named
P. exocseti, specimens of which, burrowing into the abdominal cavity of that fish, have
been recently described, 1901, by Mr Andrew Scott.
Oken classed the Lernaeidse amongst the Mollusca, removed these parasites from the
Sea Pens, Pennatula, and placed them in a distinct genus, Pennella, whilst
De Blainville suggested Lerneopenna as the generic name. Cuvier and naturalists
generally had adopted Oken's term, though some preferred the spelling Penella.
Additional species were discovered from time to time. Chamisso and Eysenhardt
described Penella diodontis from the branchiae of Diodontis mola, captured in the
Pacific ; Dekay named P. sagitta as adhering to Diodon pilosus, and von Nordmann,
in his description of P. sagitta from Lophius marmoratus, thought that it and Dekay's
specimen were the same species. Angus found a parasite on a species of Coryphwna
near the gills, which William Baird named Penella pustulosa. Milne Edwards
stated that Pennella sultana had been found in the mouth of Carenx ascensofius.
Steenstrup and Lutken gave an account of P. varians which infested a "Dolphin,"
the species of which was not determined. E. Perceval Wright described Pennella
orthagorisci from specimens obtained from Orthagoriscus mola caught in Cork har
bour in 1869. They were implanted in the skin on either side of the dorsal fin, and the
total length of the parasite from the head to the anal opening was 7 inches. He
* Named after the Swedish Councillor, Magnus Lagerstrom.
ON PEN NELL A BALDEN OPT ERA:. 411
also stated that Baird referred a Pennella from a sunfish captured in Cornwall to
P. filosa. G. M. Thomson gave an account (1889) of a Pennella found on a swordfish
(Histiophorus herschelii), which he named P. histiophori. Eamsay H. Traquair has
called my attention .to two specimens of Pennella in the Collection of the Royal Scottish
Museum, which he had provisionally named P. orihagorisci. Possibly they may have been
included in the Natural History Museum of the University, which was transferred many
years ago to the Royal Scottish Museum, but nothing definite is known of the animal
on which they were parasitic, or when they were obtained. One specimen was deprived
of the head and arms ; the other had a head and two lateral arms, but no dorsal arm,
and it was about 5 inches long.*
Observations on the Lernaeidae during the first quarter of the last century induced
naturalists to consider that these parasites were not to be regarded as Worms, Molluscs,
or Zoophytes, but that they had an affinity to the Crustacea. Their position was
finally adjusted in 1832 by Alexander von Nordmann, who, from the young having
the nonparasitic character of Cyclops, from the segmented structure of the male, which
is a free swimming animal, though it may become attached to the female, and from the
position and characters of the feet, definitely placed these curious animals amongst the
Crustacea, in which they are now generally regarded by naturalists as forming a family
of parasitic Copepoda.
An important extension of our knowledge of the hosts to which different species of
Pennella may become attached was made when it was ascertained that specimens had
been obtained imbedded in the skin of species of whales frequenting the North Atlantic
Ocean. Steenstrtjp and Lutken published in 1861 a memoir in which a Pennella was
described as attached to a Hyperoodon rostratus captured in 1855 south of the
Faroe Islands ; they named the parasite Pennella crassicornis. They referred to an
observation made some years previously by von Duben that a Pennella, species not
named, had been obtained from a Finner whale. In 1866 G. 0. Sars stated that
specimens of a Pennella with the head buried in the blubber were seen attached
to Balsenoptera musculus. In 1877 Koren and Danielssen published a memoir on a
Pennella found on Balsenoptera rostrata, and preserved in the museum at Bergen,
which they had named Pennella balsenopterse twenty years previously. Other
specimens from B. rostrata, buried with the head and hornlike arms in the blubber in
the vicinity of the external organs of generation, had subsequently been added to this
museum. Van Beneden, in his memoirs on the natural history of the Cetacea, referred to
these Balgenopterae as serving as hosts for a Pennella ; and he further stated, though
without giving very definite authorities, that this parasitic crustacean had also been
found on Balsenoptera sibbaldii, and probably on B. borealis.
* Dr Traquair showed at the meeting of the Royal Society at which this memoir was read two dried
specimens of Pennella exocseti, which he had received in November 1904 from Captain Pater. It appears that when
Captain Pater was on a voyage in the South Pacific a flying fish flew on to the ship ; and deeply rooted in the wall of
its abdomen, behind the pectoral, fin, were the two specimens of Pennella, which he removed and sent to the Royal
Scottish Museum.
412
SIR WILLIAM TURNER
As the memoir of Koren and Danielssen contains a description of the external
characters with observations on the internal anatomy of the female Pennella
balxnopterae, and is illustrated by a plate with nine figures. I have made a careful
comparison of my specimens with their description and drawings.
External Characters of the Female.
As the specimens in my possession, like those studied by Koren and Danielssen,
were not uniform in length, I have measured the longest and the shortest in order to
show the variation, and in the following table I have recorded their chief dimensions,
alongside of the corresponding measurements of two of the specimens described by the
Norwegian naturalists.
K.
&D.
Turner.
A.
B.
A.
B.
Whole length of parasite
320 mm.
300
294
mm. 206
Length of head
7
n
6
5
ii 4
Breadth of head
8
ii
7
5
ii 4
Longest hornlike arm
15
ii
14
33
.1 23
Greatest thickness of arm
2
M
2
3
3
Length of thoracicoabdominal part .
315
11
294
289
ii 202
Greatest thickness of same
6
II
6
45
4
Length of pennated abdominal part .
45
II
42
30
,. 25
It is obvious from these measurements that the females varied considerably in
length ; and as my shortest specimen had a pair of long ova strings attached to the
ventral surface, it may be assumed to be adult equally with the longest. It will be
noticed that neither of the two specimens is so long as the shortest of those recorded by
Koren and Danielssen, whilst their longest specimen was 320 mm. (12^ inches).
P. bolsenopterse is therefore a giant amongst the Copepoda.
The head, both in length and breadth, slightly smaller than in their examples, had
a stunted, clubshaped appearance. Its colour, that of the arms and of the upper part
of the socalled thoracic region, was brownishyellow, whilst the lower part of that
region and the entire extent of the abdomen was of a dark purplish hue with a shade
of green, even after the specimens had been for several months in spirit. The head,
arms, and upper part of the thorax were imbedded in the skin and blubber, on the
juices in which the parasite lived. The greenishpurpletinted part of the body floated
in the seawater, and was more or less in contact with the skin of the whale. Seen
through the medium of the water, it would approximate to the colour of the skin, and
would furnish an example of protective mimicry.
The summit of the head was studded with numerous shallow, papillalike tubercles ;
they also surrounded the cleftlike opening of the mouth, which formed a deep
mesial groove extending for a small distance on the ventral surface of the head. A
short groove was present on the dorsal surface, which had, at its upper end, a blunt,
hooklike tubercle at each margin, but in no instance did I see a pair of pointed, claw
ON PEN NELL A BAL^ENOPTEK^E. 413
like antennae, relics of the free Cyclops stage of development, such as are represented
by Koren and Danielssen in their figure 9, tab. xvi.
From the base of the head three hornlike arms arose, which extended almost
horizontally outwards ; they were the anchors of attachment implanted in the blubber
of the whale. One sprang from the mesial dorsal surface, whilst the others were right
and left lateral. They varied in length in the same specimen, and the dorsal arm was
usually the shortest. They differed also in thickness and were irregular on the surface ;
the free end was blunt (Plate I. figs. 1, 2), and in one specimen a lateral arm was
bifurcated.
The body of the parasite extended from the base of attachment of the arms to the
free end of the pennated portion. It varied materially in thickness in different parts
of its length. Immediately below the arms its transverse diameter was 3 to 4 mm. ;
it was somewhat flattened on both dorsal and ventral surfaces, and on the ventral
surface, close to the mesial line, most of the specimens showed pairs of appendages.
They were so minute as to be scarcely visible to the naked eye. In two specimens
four pairs were seen, as had been figured by Koren and Danielssen. In others, two
pairs, or even a single pair, only were recognised, and in a few they were not visible.
Their recognition was assisted by the presence of a spot of dark pigment. Four is
without doubt the typical number of these feetlike appendages, though it would seem
as if this number was not always preserved in the process of transformation from the
embryonic cyclopoid form to their retrograde condition in the adult (Plate I. fig. 2).
Eight mm. from the base of the arms the transverse diameter of the body diminished
to 1"5 mm., and for a considerable distance it preserved this diameter ; it was cylindrical
in shape, smooth on the surface, and not unlike in form and colour a steel knitting
needle. It was an elongated necklike division of the body, very characteristic of the
parasite, and may be regarded as the thoracic segment.
The body was prolonged into the abdomen, which increased in bulk, measured
4 mm. in breadth, lost its smooth appearance, and was marked by numerous transverse
constrictions, between which minute beadlike projections were arranged in rows.
The abdomen was the widest and most deeply coloured part of the body ; as it con
tained both alimentary canal and the female genital organs, it may appropriately be
named the genitoabdominal segment. At the lower end two genital openings were
seen on the ventral surface, from which depended the pair of ova strings. Immediately
above these openings was a small rounded eminence, to which probably the male
parasite may attach itself when engaged in impregnation.
The ova strings were a pair of very slender threads, yellowishbrown in colour, and
of remarkable length; in one parasite each string measured 400 mm. (157 in.). They
floated free in the surrounding medium ; they were sometimes almost straight, but at
others they had an undulating character.
The terminal part of the body was prolonged behind the genital openings from 25
to 30 mm., varying in the different specimens ; it was only 2 mm. in transverse diameter
414 SIR WILLIAM TURNER
and came to a free end. It had a caudate appearance ; but as it contained the intestinal
end of the alimentary canal, it should be regarded as the caudate segment of the
abdomen. The anal orifice was situated in a cleft at its free end. Its dorsal surface
was marked by transverse constrictions, and from its ventral surface a number of
bristlelike structures arose, which gave to the terminal part of the body the pennate
character which has decided the generic name.
Chitinous Coat.
The chitinous coat of the parasite was translucent, firm, and so tough as to turn
the edge of the razor. It was for the most part homogeneous throughout its substance,
but in places delicate lines, parallel to each other and to the plane of the surface, gave
it a laminated appearance, as if it had been formed by superposition of layers. It
varied in thickness in different regions, as was seen both in longitudinal and transverse
sections. In the head, this coat was about ^rd of a mm. thick, but at the origin of the
arms it was about f rds of a mm. In the arm itself the thickness varied in different parts.
In proximity to the head it formed about f rds of the diameter of a transverse section, in
the middle of the arm about ^rd, and near the free end about \. In the attenuated
thoracic region the proportion was about \, in the genitoabdominal part it was less,
and it was a little thinner on the ventral than on the dorsal aspect. In the pennated
abdominocaudate segment it represented about ^rd of the transverse diameter of the
parasite.
On the outer surface of the chitinous envelope a layer of cuticle was present, which
was usually closely adherent to the chitin, but in places it was partially detached, and
had probably been drawn off in cutting the sections. When examined microscopically
it was seen to be striated in a direction perpendicular to the plane of the surface ;
higher magnification showed this appearance to be due to short columnar cells, which
were arranged parallel to each other. In sections where the displaced cuticle had been
turned over so as to expose its free surface, the broader ends of the columnar cells
were seen to be at that surface, and by their close apposition to each other to form a
continuous layer.
The chitinous wall was lined by a membrane, which in various localities, to be sub
sequently referred to, was richly pigmented (figs. 17, 24, 26).
In the papillalike tubercles, in the parts of the head not occupied by the muscles,
in the thickened part of the body immediately below the head and in the arms,
an areolated tissue was situated within the membranous lining of the wall of chitin.
Structure of the Head.
The internal structure of the head was examined in a series of transverse and
longitudinal sections from its summit to the base of attachment of the arms. The
ON PEN N EL LA BALJINOPTER^E. 415
papillalike tubercles formed the most marked feature of the summit. Each had
a definite chitinous envelope, which inclosed an areolated tissue, the areolae of
which varied materially in size, and corresponded in character with the tissue in
the axis of the arms to be subsequently described.
Within the tuberculated summit numerous transversely striped muscular fibres occu
pied a large proportion of the space dorsally and laterally inclosed by the chitinous
envelope. They arose from the inner surface of the envelope, which in transverse sec
tion had a ridge and furrowlike character. The muscular fibres in this region situated
laterally to the mesial plane converged from their origin and seemed to end in
a common tendon, which was attached to the papilla like tubercles situated on the
side of the cleft which formed the oral aperture (figs. 5, 7, 8). Their apparent
function was to draw the sides of the cleft asunder, widen the aperture, and by
successive contractions and relaxations to convert the cleft into a suctorial mouth.
In transverse sections of the head below the tubercles the muscular fibres were
less numerous ; those situated in proximity to the mesial plane converged on the
dorsal wall of the alimentary canal, on which they could act directly as dilators.
The fibres situated further from the mesial plane reached the dorsal aspect of a
pair of bodies, to be immediately described, which stained readily with carmine.
The striped muscular fibres were seen as low down as the origin of the arms,
but they were absent immediately below these appendages, and their place was to
a large extent taken by the areolated tissue.
I have more than once referred to a tissue, which I have named ' areolated,' situated
in the head, in the part of the body immediately below the head, and in the arms
into which it was prolonged at their base of attachment. In a subsequent section I
shall have to call attention to a similar tissue in the abdomen. Koren and
Danielssen described a layer of adipose matter, in most places not very thick,
though it could form isolated fatty agglomerations ; in the head, arms and the upper
thoracic division of the body it formed a thick stuffing, and corresponded in its
position to the areolated tissue seen in my specimens : the adipose tissue was composed
of fat cells, which, they say, had one or more ramifications on the cell.
In its general characters the areolated tissue consisted of a meshwork of connective
tissue, continuous with the membranous lining of the chitinous wall of the parasite.
In the strands of this meshwork, more especially in its peripheral part, nucleated
cells were seen in places in considerable numbers, which in size and general
appearance were not unlike leucocytes. The areolae of the meshwork varied in
size, the largest being just visible to the naked eye, whilst the smallest required
a magnification of two hundred to three hundred diameters. In specimens taken
, from the head, when the tissue was teased with needles and examined in glycerine,
, the areolae were seen to contain rounded or ovoid cells, which, like fat cells, refracted
the light strongly, and showed the characteristic reaction of fat with osmic acid ;
, in the act of teasing, many of the fat cells were ruptured and oil globules escaped. In
416 SIR WILLIAM TURNER
sections through the head and arms, which had been treated with nitric acid in
order to soften the chitin previous to making the section, subsequently soaked in
alcohol, and then mounted in Canada balsam, the tissue was modified in appearance.
Although some of the cells retained the ovoid form and to some extent the refracting
character, the majority had more or less irregular outlines, and their contents had
generally the appearance of a granular cellplasm, not usually staining strongly
with carmine ; though sometimes the granules were relatively large, and stained
more deeply with carmine, as if they had a nuclear character. It would seem
as if, with the disappearance of the fat, the cellplasm had come into view.
In certain localities the areolated tissue showed characters deserving of more
detailed notice. In the arms, where they adjoined the head and where the areolated
tissue was small in proportion to the thickness of the arm, two large areolae, each
containing granular cellplasm with a nucleus, were very distinct (fig. 11). About
the middle of the arm, also, a pair of areolae, containing granular cellplasm, similar in
size and in close relation to the wall, were present ; but as the areolated tissue in this
part of the arm was much more abundant than near the head, a cluster of large areolae
also occupied the central area of the tissue (fig. 12). A somewhat similar appearance
was seen in the relatively smaller amount of this tissue near the tip of the arm.
In some sections the areolated tissue in the arms was modified in a peculiar
manner. Whilst in some of the areolae the refracting character of fat cells was dis
tinctive, many others, especially those of large size, were crowded with nuclei, which
stained deeply with carmine. The nuclei were so closely set that the amount of cell
plasm associated with each nucleus was extremely small, and the latter dominated in
quantity and distinctness over the cellplasm. It seemed as if an extensive proliferation
of the nuclei had taken place (fig. 13).
In sections through the head in proximity to the arms, where the areolated
tissue was relatively abundant, the largest areolae with their contained cells occupied
the midarea of the tissue, whilst the smaller areolae formed its peripheral part (fig. 9).
The tissue which constituted the axis of the papillalike tubercles of the head con
sisted of the smaller type of areolae, though they were not uniform in size, as some
were four or five times larger than others.
It should be noted that the part of the parasite immediately below the arms
had on the ventral surface the pairs of limblike appendages already referred to.
They were so extremely rudimentary that it was difficult to recognise them with the
naked eye, and sometimes even they were absent. It is within this part of the body
that the areolated tissue was most abundant. Had the limbs been functionally active,
one cannot doubt but that an adequate amount of striped muscle would have been
developed in this region as their motor apparatus ; but, under the changed conditions,
it was no longer required, and its place had been taken by a passive, areolated tissue
containing fat cells.
In addition to the oesophagus, the muscular fibres, and the areolated tissue, the
ON PENNELLA BALJENOPTERJE. 417"
chitinous wall of the head inclosed three objects — a pair placed laterally, which
were readily coloured by carmine (fig. 8, g), and one placed mesially next the ventral
surface, which did not take the carmine dye (fig. 8, V).
The red stained bodies were recognised in sections through the head as high
as the sides of the oral chink, and were obviously nerve ganglia. At their upper
end they were separated from each other by the mesial oral chink, the tubercles
connected with its walls and the areolated tissue associated with the tubercles.
Each was placed close to the common tendon of attachment of the bundle of striped
muscular fibres already described on each side of the head. In the upper part of
a ganglion not more than six to twelve characteristic cells could be seen in the
plane of section, but opposite the lower end of the oral chink the ganglion increased
in size and the cells were much more numerous. Immediately below the oral cleft
the ganglia were relatively large, and were situated partly to the side of the oesophagus
and partly ventrally to it, but they were not continuous with each other on the
ventral surface, as they were separated by the mid ventral object which did not
take the carmine stain. The ganglia were traced in successive sections as far as
opposite the origins of the arms, but they were not visible in the sections immediately
below the arms, where their place was occupied by areolated tissue. It was noticed
that where each ganglion had a wide transverse diameter, it was not unusual for the
cells in its centre to show signs of disintegration ; and sometimes this was so ex
tensive that a cavity had formed, the wall of which was irregular and showed no sign
of a lining membrane (fig. 8, g, g).
When examined under a high magnifying power the structure of the ganglion cells
was readily recognised. The nuclei were large and oval in shape, and as they stained
a deep red with carmine, they were very distinct, and an intranuclear network of
fibrillse was present in them. The cellplasm was granulated. The bestmarked cells
were considerably larger than the motor cells in the lumbar enlargement of the human
spinal cord, though others were very much smaller. The bodies of the cells were
polygonal, and from the angles delicate processes of the cellplasm projected. As a
rule, the cells were closely aggregated, and it was difficult to trace these processes for
any distance, but they were sufficiently distinct to leave no doubt of the multipolar
character of the cells. In places minute intercellular intervals were visible, and the
outlines of the cells were defined by a distinct wall. Although the relative proportion
of the nucleus to the cellplasm varied in the cells, it was evident that in the largest
cells the cellplasm exceeded three or even four times in quantity the size of
the nucleus (fig. 15).
From the character of the cells there can, I think, be no doubt that the red stained
bodies were a pair of nerve ganglia. Their position in the head, their relation to its
ventral surface and to the oesophagus, localise them as oesophageal ganglia, situated
laterally and ventrally to the gullet, though not united to each other on the ventral
aspect of the oesophagus. When portions of these ganglia were removed, teased with
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 18). 62
418 SIR WILLIAM TURNER
needles, and stained with picrocarmine, delicate fibres were seen to lie between the cells
•and to emerge from the ganglia, which, from their association with the nerve cells,
were obviously nerve fibres. Some were nonmedullated ; others, again, apparently
contained a medullary substance, which had aggregated into little clumps within
the neurilemma.
The midventral object above referred to, when examined in relatively thick sections
and under low magnification, seemed to be a solid cordlike body, lying mesially in the
long axis of the head. It was situated between the ventral aspect of the ali
mentary canal and the inner surface of the ventral chitinous wall of the head.
In longitudinal sections it was traced as high as the muscles of the head, the fibres of
which arched above it, from their origin from the envelope of chitin to the side of the
oral cleft. Its transverse diameter was greater than the anteroposterior, and it was
bounded by a distinct capsule of fibrous tissue, which gave it a definite outline and
differentiated it from the surrounding structures. The oesophageal ganglia were in
relation to its sides, and in places even encroached on its ventral surface, and their
upper ends were in the same transverse plane as the upper limit of its investing capsule.
Below the ganglia it was bounded by the areolated tissue which was so abundant at
and immediately below the arms. From its position it might have been taken for an
axial nerve cord associated with the oesophageal ganglia, but no fibres could be
detected in it, and it did not stain with carmine (fig. 8, V).
When thin sections were examined with a Zeiss lens x 250 the capsule was seen to be
lined by a layer of rounded cells ; in favourable sections they formed a continuous lining,
but not unfrequently they were arranged in patches, separated by intervals. The cells
were much smaller than the nuclei of the nerve cells in the adjoining ganglia, they were
nucleated, and the cellplasm was dimly granular. The material generally inclosed by
the capsule had a granular character, and, as a rule, showed no trace of structure, and
was possibly a coagulated substance. Sometimes, however, nucleated cells of great
translucency were interspersed in the granular material, and fattylooking globules were
occasionally present.
In sections through the body of the parasite in the thoracic segment the corre
sponding arrangement, interposed between the alimentary canal and the ventral
wall of chitin, was the ventral mesial space, so that the midventral object above
described was obviously a prolongation upwards into the head of the ventral space
of the coelom.
In some of the transverse sections through the parasite made a little above the
attachment of the arms a special appearance was seen. It consisted in the presence
of a band or column of chitin, almost circular in outline, lying in relation to the dorsal
space and interposed between the oesophagus and the inner surface of the dorsal wall
of the chitinous envelope, and apparently quite independent of it. It was difficult to
give a satisfactory explanation of the part which the band played in the economy of
the parasite (figs. 8, 10, Ch).
ON PEN N ELLA BAL&NOPTERJE 419"
Alimentary Canal.
The canal extended in a direct line from mouth to anus, and had no convolutions in
any part of its course. The oral cleft passed deeply into the substance of the ventral
surface of the head, and was continued at its lower part into a relatively wide oesophagus,
down which a bristle could readily be passed.
In transverse sections through the upper part of the oesophagus, the diameter from
side to side was seen to be much greater than in the dorsiventral direction, and the
opposite walls were almost in contact. The ventral wall of the canal was in close rela
tion with the capsule of the mid ventral space of the coelom, which lay between it and
the chitinous wall of the head, the dorsal wall was in relation to the musculature of the
head, and the sides were in contact with the oesophageal ganglia (fig. 8).
In the lower part of the head, where the muscular fibres were replaced by areolated
tissue, the dorsal wall of the canal was separated from the chitinous envelope by the
dorsal space, which contained a granulated material, possibly a coagulum. The space
was bounded by a fibrous membrane, which was lined by nucleated cells, though
frequently they were in patches and did not form a continuous layer. These cells were
about the size of leucocytes, and not unlike them in appearance. The muscular wall of
the alimentary canal was attached to the areolated tissue at its sides by bands, formed
of connective tissue and nonstriped muscle, which constituted short lateral mesenteries ;
between these bands were narrow channels, in which blood or other nutritive fluid may
have circulated.
Transverse sections through the body immediately below the arms showed the
alimentary canal in the axis of the section, with a space in relation to both its dorsal
and ventral surfaces. The lumen of the canal was not so compressed dorsiventrally as
in the head. Wellmarked areolated tissue surrounded the canal with its dorsal and
ventral spaces, and closely packed the whole area between them and the inner surface of
the chitinous wall (fig. 9). As it efficiently supported the canal, the lateral mesenteries
were short and their fibres were continued into the meshwork of the areolae, which
again was continuous with the membrane lining the inner surface of the wall. A few
scattered pigment cells were seen in this membrane, though not nearly so abundant
as lower down in the thoracic segment of the body.
In sections through the attenuated thoracic segment the areolated tissue was
no longer present, and the space inclosed by the chitinous wall was occupied by the
alimentary canal and the dorsal and ventral spaces. The canal was in the axis of the
section and was reniform in shape ; its lateral angles were in such close relation to the
lining membrane of the chitin that the mesenteries were practically absent (fig. 16). The
dorsal and ventral spaces were proportionally large, almost equal in size, and were situ
ated between the lining membrane and the corresponding wall of the alimentary canal.
Each space was inclosed by a definite wall of fibrous membrane, the inner surface of
which was lined by a layer of nucleated cells ; the cellplasm in some was granular
420 SIR WILLIAM TURNER
in character, though in others it was more translucent. The spaces were frequently
devoid of contents, though in some sections irregular fragments, granular in appearance
and possibly a coagulated substance, were present. The dorsal and ventral spaces, not
only in relation to this, but to other divisions of the alimentary canal, formed the
ccelom or body cavity. Koren and Danielssen named the dorsal space the dorsal
canal, and stated that during life it was full of red thinly flowing blood.
The chitinous wall was lined by a definite membrane, in which was a layer of large
stellate cells, full of a rich purplishblack pigment.
The alimentary canal and the associated spaces retained the characters just described
as far down the body as where the attenuated thoracic part was continued into the
genitoabdominal segment, in which the chitinous wall also possessed a lining
membrane with large richlypigmented cells. The alimentary canal was in the axis
of the segment, and its transverse section was almost round, and so capacious that it
may properly be regarded as the stomach. Each lateral aspect was attached to the
adjoining pigmented membrane by a mesentery. The dorsal and ventral spaces were
relatively small. Between the canal a